TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES IV: DISCRETENESS AND SECTIONS YUICHIRO HOSHI AND SHINICHI MOCHIZUKI JULY 2022 Abstract. Let Σ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardi- nality one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated config- uration spaces over algebraically closed fields in which the primes of Σ are invertible. The present paper focuses on the topic of compar- ison between the theory developed in earlier papers concerning pro- Σ fundamental groups and various discrete versions of this theory. We begin by developing a theory of combinatorial analogues of the section conjecture and Grothendieck conjecture in anabelian geometry for abstract combinatorial versions of the data that arises from a hyperbolic curve over a complete discretely valued field, under the condition that, for some l ∈ Σ, the l-adic cyclotomic character has infinite image. This portion of the theory is purely combina- torial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs [satisfying certain conditions] — a result which may be regarded as a geomet- ric interpretation of the well-known elementary fact that free pro-Σ groups are torsion-free. We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hy- perbolic topological surfaces to the effect that various properties of [discrete] subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under considera- tion. These results make possible a fairly straightforward trans- lation, into discrete versions, of pro-Σ results obtained in previous papers by the authors concerning the theory of partial combinatorial cuspidalization, Dehn multi-twists, the tripod homomorphism, metric- admissibility, and the characterization of local Galois groups in the global Galois image associated to a hyperbolic curve. Finally, we con- sider the analogue of the theory of tripods [i.e., copies of the pro- Σ or discrete fundamental group of the projective line minus three points] associated to cycles in a hyperbolic topological surface. From an intuitive topological point of view, these tripods are obtained by 2010 Mathematics Subject Classification. Primary 14H30; Secondary 14H10. Key words and phrases. anabelian geometry, combinatorial anabelian geometry, combinatorial section conjecture, fixed points, combinatorial Grothendieck conjec- ture, combinatorial cuspidalization, discrete/profinite comparison, liftings of cycles, tripods, semi-graph of anabelioids, semi-graph of temperoids, hyperbolic curve, con- figuration space. 1 2 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI considering once-punctured tubular neighborhoods of the cy- cles. Such a construction was considered previously by M. Boggi in the discrete case, but in the present paper, we consider it from the point of view of the abstract pro-Σ theory developed in earlier pa- pers by the authors and then proceed to relate this theory to the discrete theory by applying the tools developed in earlier portions of the present paper. Contents Introduction 2 0. Notations and Conventions 12 1. The combinatorial section conjecture 13 2. Discrete combinatorial anabelian geometry 46 3. Canonical liftings of cycles 94 Appendix. Explicit limit seminorms associated to sequences of toric surfaces 120 References 130 Introduction Let Σ ⊆ Primes be a subset of the set of prime numbers Primes which is either equal to Primes or of cardinality one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated configuration spaces over al- gebraically closed fields in which the primes of Σ are invertible [cf. [CmbGC], [MT], [CmbCsp], [NodNon], [CbTpI], [CbTpII], [CbTpIII]]. The present paper focuses on the topic of understanding the relation- ship between the theory developed in earlier papers concerning pro-Σ fundamental groups and various discrete versions of this theory. This topic of comparison of pro-Σ and discrete versions of the theory turns out to be closely related, in many situations, to the theory of sections of various natural surjections of profinite groups. Indeed, this rela- tionship with the theory of sections is, in some sense, not surprising, inasmuch as sections typically amount to some sort of fixed point within a profinite continuum. That is to say, such fixed points are often closely related to the identification of a rigid discrete structure within the profinite continuum. In §1, §2, we study two different aspects of this topic of compari- son of pro-Σ and discrete structures. Both §1 and §2 follow the same pattern: we begin by proving an abstract and somewhat technical com- binatorial result and then proceed to discuss various applications of this combinatorial result. In §1, the main technical combinatorial result is summarized in The- orem A below [where Σ is allowed to be an arbitrary nonempty set of COMBINATORIAL ANABELIAN TOPICS IV 3 prime numbers]. This result consists of versions of the section con- jecture and Grothendieck conjecture — i.e., the central issues of concern in anabelian geometry — for outer representations of ENN-type [cf. Definition 1.7, (i)]. Here, we remark that outer repre- sentations of ENN-type are generalizations of the outer representations of NN-type studied in [NodNon]. Just as an outer representation of NN-type may be described, roughly speaking, as a purely combinato- rial object modeled on the outer Galois representation arising from a hyperbolic curve over a complete discretely valued field whose residue field is separably closed, an outer representation of ENN-type may be described, again roughly speaking, as an analogous sort of purely com- binatorial object that arises in the case where the residue field is not necessarily separably closed. The pro-Σ section conjecture portion of Theorem A [i.e., Theorem 1.13, (i)] is then obtained by combining • the essential uniqueness of fixed points of certain group actions on profinite graphs given in [NodNon], Proposition 3.9, (i), (ii), (iii), with • an essentially classical result concerning the existence of fixed points [cf. Lemma 1.6; Remarks 1.6.1, 1.6.2], which amounts, in essence, to a geometric reformulation of the well-known fact that free pro-Σ groups are torsion-free [cf. Remarks 1.13.1; 1.15.2, (i)]. The argument applied to prove this pro-Σ section conjecture portion of Theorem A is essentially similar to the argument applied in the tem- pered case discussed in [SemiAn], Theorems 3.7, 5.4, which is reviewed [in slightly greater generality] in the tempered section conjecture por- tion of Theorem A [cf. Theorem 1.13, (ii)]. These section conjecture portions of Theorem A imply, under suitable conditions, that there is a natural bijection between conjugacy classes of pro-Σ and tempered sections [cf. Theorem 1.13, (iii)]. This implication may be regarded as an important example of the phenomenon discussed above, i.e., that considerations concerning sections are closely related to the topic of comparison of pro-Σ and discrete structures. Finally, by combining the pro-Σ section conjecture portion of Theorem A with the combinatorial version of the Grothendieck conjecture obtained in [CbTpII], Theorem 1.9, (i), one obtains the Grothendieck conjecture portion of Theorem A [cf. Corollary 1.14]. Theorem A (Combinatorial versions of the section conjecture and Grothendieck conjecture). Let Σ be a nonempty set of prime numbers, G a semi-graph of anabelioids of pro-Σ PSC-type, G a profi- nite group, and ρ : G → Aut(G) a continuous homomorphism that is of ENN-type for a conducting subgroup I G ⊆ G [cf. Definition 1.7, (i)]. Write Π G for the [pro-Σ] fundamental group of G and Π tp G for the 4 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI tempered fundamental group of G [cf. [SemiAn], Example 2.10; the dis- cussion preceding [SemiAn], Proposition 3.6]. [Thus, we have a natural outer injection Π tp G → Π G — cf. [CbTpIII], Lemma 3.2, (i); the proof of def out [CbTpIII], Proposition 3.3, (i), (ii).] Write Π G = Π G  G [cf. the dis- def out tp cussion entitled “Topological groups” in [CbTpI], §0]; Π tp G = Π G  G; G  → G, G  tp → G for the universal pro-Σ and pro-tempered coverings of G corresponding to Π G , Π tp G ; VCN(−) for the set of vertices, cusps, and nodes of the underlying [pro-]semi-graph of a [pro-]semi-graph of an- abelioids [cf. Definition 1.1, (i)]. Thus, we have a natural commutative diagram tp 1 −−−→ Π tp G −−−→ Π G −−−→ G −−−→ 1 ⏐ ⏐  ⏐ ⏐     1 −−−→ Π G −−−→ Π G −−−→ G −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows  tp are outer injections; Π tp G acts naturally on G ; Π G acts naturally on  Then the following hold: G. (i) Suppose that ρ is l-cyclotomically full [cf. Definition 1.7, (ii)] for some l ∈ Σ. Let s : G → Π G be a continuous section of the natural surjection Π G  G. Then, relative to the action of  via conjugation of VCN-subgroups, the image Π G on VCN( G)  of s stabilizes some element of VCN( G). tp (ii) Let s : G → Π G be a continuous section of the natural surjec- tp  tp tion Π tp G  G. Then, relative to the action of Π G on VCN( G ) via conjugation of VCN-subgroups [cf. Definition 1.9], the im- age of s stabilizes some element of VCN( G  tp ). (iii) Write Sect(Π G /G) for the set of Π G -conjugacy classes of con- tinuous sections of the natural surjective homomorphism Π G  tp G and Sect(Π tp G /G) for the set of Π G -conjugacy classes of continuous sections of the natural surjective homomorphism Π tp G  G. Then the natural map Sect(Π tp G /G) −→ Sect(Π G /G) is injective. If, moreover, ρ is l-cyclotomically full for some l ∈ Σ, then this map is bijective. (iv) Let H be a semi-graph of anabelioids of pro-Σ PSC-type, H a profinite group, ρ H : H → Aut(H) a continuous homomor- phism that is of ENN-type for a conducting subgroup I H ⊆ H. Write Π H for the [pro-Σ] fundamental group of H. Suppose further that ρ is verticially quasi-split [cf. Defini- ∼ tion 1.7, (i)]. Let β : G → H be a continuous isomorphism def such that β(I G ) = I H ; l ∈ Σ a prime number such that ρ G = ρ COMBINATORIAL ANABELIAN TOPICS IV 5 ∼ and ρ H are l-cyclotomically full; α : Π G → Π H a continuous isomorphism such that the diagram G ρ G −→ Aut(G) −→ Out(Π G ) | | ↓ β ↓ H ρ H −→ Aut(H) −→ Out(Π H ) — where the right-hand vertical arrow is the isomorphism ob- tained by conjugating by α — commutes. Then α is graphic [cf. [CmbGC], Definition 1.4, (i)]. The purely combinatorial theory of §1 — i.e., the theory surrounding and including Theorem A — has important applications to scheme theory — i.e., to the theory of hyperbolic curves over quite general complete discretely valued fields — as follows: (A-1) We observe that a quite general result in the style of the main results of [PS] concerning valuations fixed by sections of the arithmetic fundamental group follows formally, in the case of hyperbolic curves over quite general complete discretely valued fields, from Theorem A [cf. Corollary 1.15, (iii); Re- mark 1.15.2, (i), (ii)]. The quite substantial generality of this result is a reflection of the purely combinatorial nature of Theorem A. This approach contrasts substantially with the approach of [PS] via essentially scheme-theoretic techniques such as the local-global principle for the Brauer group [cf. Re- mark 1.15.2, (i)]. The approach of the present paper also differs substantially from [PS] in that the transition from fixed points of graphs to fixed valuations is treated as a formal consequence of well-known elementary properties of Berkovich spaces, i.e., in essence the compactness of the unit interval [0, 1] ⊆ R [cf. Remark 1.15.2, (ii)]. (A-2) We observe that the natural bijection between conjugacy classes of pro-Σ and tempered sections discussed in the purely com- binatorial setting of Theorem A implies a similar bijection in the case of hyperbolic curves over quite general complete dis- cretely valued fields [cf. Corollary 1.15, (vi)]. This portion of the theory was partially motivated by discussions between the second author and Y. André. In the context of (A-1), we remark that, in the Appendix to the present paper, we give an elementary exposition from the point of view of two- dimensional log regular log schemes of the phenomenon of conver- gence of valuations, without applying the language or notions, such 6 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI as Stone-Čech compactifications, typically applied in expositions of the theory of Berkovich spaces. In §2, we turn to the task of formulating discrete analogues of a substantial portion of the theory developed in earlier papers. This formulation centers around the notion of a semi-graph of temper- oids of HSD-type [i.e., “hyperbolic surface decomposition type” — cf. Definition 2.3, (iii)], which may be thought of as a natural discrete analogue of the notion of a semi-graph of anabelioids of pro-Σ PSC- type [cf. [CmbGC], Definition 1.1, (i)]. As the name suggests, this notion may be thought of as referring to the sort of collection of dis- crete combinatorial data that one may associate to a decomposition of a hyperbolic surface into hyperbolic subsurfaces. Alternatively, it may be thought of as referring to the sort of collection of combinatorial data that arises from systems of topological coverings of the system of topological spaces naturally associated to a stable log curve over a log point whose underlying scheme is the spectrum of the field of complex numbers [cf. Example 2.4, (i)]. After discussing various basic proper- ties and terms related to semi-graphs of temperoids of HSD-type [cf. Proposition 2.5; Definitions 2.6, 2.7], we observe that the fundamen- tal operations of restriction, partial compactification, resolution, and generization discussed in [CbTpI], §2, admit natural compatible analogues for semi-graphs of temperoids of HSD-type [cf. Definitions 2.8, 2.9; Proposition 2.10]. The main technical combinatorial result of §2 is summarized in The- orem B below. This result asserts, in effect, that discrete subgroups of the discrete fundamental group of a semi-graph of temperoids of HSD- type satisfy various properties of interest if and only if the profinite completions of these discrete subgroups satisfy analogous properties [cf. Theorem 2.15; Corollary 2.19, (i)]. The main technical tool that is applied in order to derive this result is the fact that any inclusion of a finitely generated group into a [finitely generated] free discrete group is, after possibly passing to a suitable finite index subgroup, necessarily split [cf. [SemiAn], Corollary 1.6, (ii), which is applied in the proof of Lemma 2.14, (i), of the present paper]. Here, we recall that in [SemiAn], this fact [i.e., [SemiAn], Corollary 1.6, (ii)] is obtained as an immedi- ate consequence of “Zariski’s main theorem for semi-graphs” [cf. [SemiAn], Theorem 1.2]. Theorem B (Profinite versus discrete subgroups). Let G, H be semi-graphs of temperoids of HSD-type [cf. Definition 2.3, (iii)].  H  for the semi-graphs of anabelioids of pro-Primes PSC- Write G, type determined by G, H [cf. Proposition 2.5, (iii), in the case where Σ = Primes], respectively; Π G , Π H for the respective fundamental groups of G, H [cf. Proposition 2.5, (i)]; Π G  , Π H  for the respective  H.  Then the following hold: [profinite] fundamental groups of G, COMBINATORIAL ANABELIAN TOPICS IV 7 (i) Let H, J ⊆ Π G be subgroups. Since Π G injects into its pro- l completion for any l ∈ Primes [cf. Remark 2.5.1], let us regard subgroups of Π G as subgroups of the profinite completion  G of Π G . Write H, J ⊆ Π  G for the closures of H, J in Π  G , Π respectively. Suppose that the following conditions are satisfied: (a) The subgroups H and J are finitely generated. (b) If J is of infinite index in Π G , then J is of infinite  G . index in Π [Here, we note that condition (b) is automatically satisfied when- ever Cusp(G)  = ∅ — cf. [SemiAn], Corollary 1.6, (ii).] Then the following hold: (1) It holds that J = J ∩ Π G .  G such that (2) Suppose that there exists an element γ  ∈ Π  −1 . H ⊆ γ  · J · γ Then there exists an element δ ∈ Π G such that H ⊆ δ · J · δ −1 . (ii) Let ∼ α : Π G −→ Π H ∼ be an outer isomorphism. Write α  : Π G  → Π H  for the outer isomorphism determined by α and the natural outer isomor- ∼ ∼  G →  H → phisms Π Π G  , Π Π H  of Proposition 2.5, (iii). Then the outer isomorphism α is group-theoretically ver- ticial (respectively, group-theoretically cuspidal; group- theoretically nodal; graphic) [cf. Definition 2.7, (i), (ii)] if and only if the outer isomorphism α  is group-theoretically verticial [cf. [CmbGC], Definition 1.4, (iv)] (respectively, group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)]; group-theoretically nodal [cf. [NodNon], Definition 1.12]; graphic [cf. [CmbGC], Definition 1.4, (i)]). The significance of Theorem B lies in the fact that it renders possi- ble a fairly straightforward translation of a substantial portion of the profinite results obtained in earlier papers by the authors into discrete versions, as follows: (B-1) the partial combinatorial cuspidalization obtained in [CbTpI], Theorem A; [CbTpII], Theorem A [cf. Corollary 2.20 of the present paper]; (B-2) the theory of Dehn multi-twists summarized in [CbTpI], Theorem B [cf. Corollary 2.21 of the present paper]; (B-3) the theory of the tripod homomorphism and metric-admissibility summarized in [CbTpII], Theorem C; [CbTpIII], Theorems A, C, D [cf. Theorem 2.24 of the present paper]; 8 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (B-4) the archimedean analogue [cf. Corollary 2.25 of the present paper] of the characterization, given in [CbTpIII], Theorem B, of nonarchimedean local Galois groups in the global Galois image associated to a hyperbolic curve. Finally, in §3, we examine the theory of canonical liftings of cy- cles discussed in [Bgg2] from the point of view of the profinite theory developed so far by the authors. This approach contrasts substan- tially with the intuitive topological approach of [Bgg2] in the discrete case. From a naive topological point of view, the canonical liftings of cycles in question amount to once-punctured tubular neighbor- hoods of the given cycles [cf. Figure 1 below], i.e., to the construction of a tripod [i.e., a copy of the projective line minus three points] canon- ically and functorially associated to the cycle. This tripod satisfies a remarkable rigidity property, i.e., it admits a canonical isomor- phism, subject to almost no indeterminacies, with a given fixed tripod that is independent of the choice of the cycle. Moreover, this canonical isomorphism is functorial with respect to “geometric” outer automor- phisms of the profinite fundamental group of the stable log curve under consideration that lift to automorphisms of the profinite fundamental group of a configuration space [associated to the stable log curve] of sufficiently high dimension. Here, by “geometric”, we mean that the outer automorphism under consideration lies in the kernel of the tripod homomorphism studied in [CbTpII], §3. Indeed, this remarkable rigid- ity property is obtained as an immediate consequence of the theory of tripod synchronization developed in [CbTpII], §3. The profinite version of the theory of canonical liftings of cycles developed in §3 is summarized in Theorem C below [cf. Theorem 3.10]. By applying the translation apparatus developed in §2 to this profinite version of the theory, we also obtain a corresponding discrete version of the theory of canonical liftings of cycles [cf. Theorem 3.14]. Theorem C (Canonical liftings of cycles). Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; Σ a set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one; k an algebraically closed field of characteristic ∈  Σ; def log def log S = Spec(k) the log scheme obtained by equipping S = Spec(k) with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over S log . For positive integers m ≤ n, write X n log for the n-th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in [CbTpI], §0]; Π n COMBINATORIAL ANABELIAN TOPICS IV 9 for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (X n log )  π 1 (S log ); log log p log n/m : X n −→ X m , p Π n/m : Π n  Π m , def Π n/m = Ker(p Π G, Π G n/m ) ⊆ Π n , for the objects defined in the discussion at the beginning of [CbTpII], §3; [CbTpII], Definition 3.1. Let I ⊆ Π 2/1 ⊆ Π 2 be a cuspidal inertia group associated to the diagonal cusp of a fiber of p log 2/1 ; Π tpd ⊆ Π 3 a 3-central {1, 2, 3}-tripod of Π 3 [cf. [CbTpII], Definition 3.7, (ii)]; I tpd ⊆ Π tpd a cuspidal subgroup of Π tpd that does not arise from a ∗ ∗∗ cusp of a fiber of p log 3/2 ; J tpd , J tpd ⊆ Π tpd cuspidal subgroups of Π tpd ∗ ∗∗ such that I tpd , J tpd , and J tpd determine three distinct Π tpd -conjugacy classes of closed subgroups of Π tpd . [Note that one verifies immediately from the various definitions involved that such cuspidal subgroups I tpd , ∗ ∗∗ , and J tpd always exist.] For positive integers n ≥ 2, m ≤ n and J tpd FC α ∈ Aut (Π n ) [cf. [CmbCsp], Definition 1.1, (ii)], write α m ∈ Aut FC (Π m ) for the automorphism of Π m determined by α; Aut FC (Π n , I) ⊆ Aut FC (Π n ) for the subgroup consisting of β ∈ Aut FC (Π n ) such that β 2 (I) = I; Aut FC (Π n ) G ⊆ Aut FC (Π n ) for the subgroup consisting of β ∈ Aut FC (Π n ) such that the image of β via the composite Aut FC (Π n )  Out FC (Π n ) → Out FC (Π 1 ) → Out(Π G ) — where the second arrow is the natural injection of [NodNon], Theo- rem B, and the third arrow is the homomorphism induced by the natural ∼ outer isomorphism Π 1 → Π G — is graphic [cf. [CmbGC], Definition 1.4, (i)]; def Aut FC (Π n , I) G = Aut FC (Π n , I) ∩ Aut FC (Π n ) G ; Cycle n (Π 1 ) for the set of n-cuspidalizable cycle-subgroups of Π 1 [cf. Defini- tion 3.5, (i), (ii)]; Tpd I (Π 2/1 ) for the set of closed subgroups T ⊆ Π 2/1 such that T is a tripodal sub- group associated to some 2-cuspidalizable cycle-subgroup of Π 1 [cf. Definition 3.6, (i)], and, moreover, I is a distinguished cuspidal subgroup [cf. Definition 3.6, (ii)] of T . Then the following hold: (i) Let n ≥ 3 be a positive integer. Then Aut FC (Π n , I) G acts naturally on Cycle n (Π 1 ), Tpd I (Π 2/1 ); there exists a unique Aut FC (Π n , I) G -equivariant map C I : Cycle n (Π 1 ) −→ Tpd I (Π 2/1 ) 10 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI such that, for every J ∈ Cycle n (Π 1 ), C I (J) is a tripodal sub- group associated to J [cf. Definition 3.6, (i)]. Moreover, there exists an assignment Cycle n (Π 1 ) J → syn I,J — where syn I,J denotes an I-conjugacy class of isomorphisms ∼ Π tpd → C I (J) — such that (a) syn I,J maps I tpd bijectively onto I, ∗ ∗∗ (b) syn I,J maps the subgroups J tpd , J tpd bijectively onto lift- ing cycle-subgroups of C I (J) [cf. Definition 3.6, (ii)], and (c) for α ∈ Aut FC (Π n , I) G , the diagram [of I tpd -, I-conjugacy classes of isomorphisms] Π tpd −−−→ ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J)  1 C I (J) −−−→ C I (α 1 (J)) — where the upper horizontal arrow is the [uniquely de- termined — cf. the commensurable terminality of I tpd in Π tpd discussed in [CmbGC], Proposition 1.2, (ii)] I tpd - conjugacy class of automorphisms of Π tpd that lifts T Π tpd (α) [cf. [CbTpII], Definition 3.19] and preserves I tpd ; the lower horizontal arrow is the I-conjugacy class of isomorphisms induced by α 2 [cf. the “Aut FC (Π n , I) G -equivariance” men- tioned above] — commutes up to possible composition with the cycle symmetry of C I (α 1 (J)) associated to I [cf. Definition 3.8]. Finally, the assignment J → syn I,J is uniquely determined, up to possible composition with cy- cle symmetries, by these conditions (a), (b), and (c). (ii) Let n ≥ 4 be a positive integer, α ∈ Aut FC (Π n , I) G , and J ∈ Cycle n (Π 1 ). Then there exists an automorphism β ∈ Aut FC (Π n , I) G such that the FC-admissible outer automorphism of Π 3 determined by β 3 lies in the kernel of the tripod homo- morphism T Π tpd of [CbTpII], Definition 3.19, and, moreover, α 1 (J) = β 1 (J). Finally, the diagram [of I tpd -, I-conjugacy classes of isomorphisms] Π tpd ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J) =syn I,β (J)  1 1 C I (J) −−−→ C I (α 1 (J)) = C I (β 1 (J)) COMBINATORIAL ANABELIAN TOPICS IV 11 — where the lower horizontal arrow is the isomorphism in- duced by β 2 [cf. the “Aut FC (Π n , I) G -equivariance” mentioned in (i)] — commutes up to possible composition with the cycle symmetry of C I (α 1 (J)) = C I (β 1 (J)) associated to I. a cycle lifting cycles Figure 1: A cycle and lifting cycles Acknowledgment The authors would like to thank Yu Iijima and Yu Yang for pointing out minor errors in an earlier version of the present paper. The first author was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science. 12 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 0. Notations and Conventions Sets: Let S be a finite set. Then we shall write S  for the cardinality of S. Let S be a set equipped with an action by a group G. Then we shall write S G ⊆ S for the subset consisting of elements of S fixed by the action of G on S. Numbers: Write Primes for the set of all prime numbers. Let Σ be a set of prime numbers. Then we shall refer to a nonzero integer n as a Σ-integer if every prime divisor of n is contained in Σ. The notation R will be used to denote the set, additive group, or field of real numbers, each of which we regard as being equipped with its usual topology. The notation C will be used to denote the set, additive group, or field of complex numbers, each of which we regard as being equipped with its usual topology. Groups: Let Σ be a set of prime numbers and f : G → H a homomor- phism (respectively, outer homomorphism) of groups. Then we shall say that f is Σ-compatible if the homomorphism (respectively, outer homomorphism) f Σ : G Σ → H Σ between pro-Σ completions induced by f is injective. Note that one verifies easily that if G is a group, and H ⊆ G is a subgroup of G of finite index, then the natural inclusion H → G is Primes-compatible. If G is a topological group, then we shall write G ab for the abelianization of G, i.e., the quotient of G by the closed normal subgroup of G generated by the commutators of G. If G is a profinite group, then we shall write G  G Σ-ab-free for the maximal pro-Σ abelian torsion-free quotient of G. We shall use the terms normally terminal and commensurably terminal as they are defined in the discussion entitled “Topological groups” in [CbTpI], §0. If I, J ⊆ G are closed subgroups of a topological group G, then we shall write I ≺ J if some open subgroup of I is contained in J. COMBINATORIAL ANABELIAN TOPICS IV 13 1. The combinatorial section conjecture In the present §1, we study outer representations of ENN-type [cf. Definition 1.7, (i), below] on the fundamental group of a semi-graph of anabelioids of PSC-type [cf. [CmbGC], Definition 1.1, (i)]. Roughly speaking, such outer representations may be thought of as an abstract combinatorial version of the natural outer representation of the maxi- mal tamely ramified quotient of the absolute Galois group of a complete local field on the logarithmic fundamental group of the geometric spe- cial fiber of a stable model of a pointed stable curve over the complete local field. By comparison to the outer representation of NN-type stud- ied in [NodNon], outer representations of ENN-type correspond to the situation in which the residue field of the complete local field under consideration is not necessarily separably closed. Such outer represen- tations of ENN-type give rise to a surjection of profinite groups, which corresponds, in the case of pointed stable curves over complete local fields, to the surjection from the arithmetic fundamental group to [some quotient of] the absolute Galois group of the base field. Our first main result [cf. Theorem 1.13, (i), below] asserts that, under the additional assumption that the associated cyclotomic character has open image, any section of this surjection necessarily admits a fixed point [i.e., a fixed vertex or edge]. This “combinatorial section conjecture” is ob- tained as an immediate consequence of an essentially classical result concerning fixed points of group actions on graphs [cf. Lemma 1.6 be- low]. By applying this existence of fixed points, we show that there is a natural bijection between conjugacy classes of profinite sections and conjugacy classes of tempered sections [cf. Theorem 1.13, (iii), below] and derive a rather strong version of the combinatorial Grothendieck conjecture [cf. [NodNon], Theorem A; [CbTpII], Theorem 1.9] for cy- clotomically full outer representations of ENN-type [cf. Corollary 1.14 below]. We also observe in passing that a generalization of the main result of [PS] may be obtained as a consequence of the theory discussed in the present §1 [cf. Corollary 1.15 below]. In the present §1, let Σ be a nonempty set of prime numbers and G a semi-graph of anabelioids of pro-Σ PSC-type [cf. [CmbGC], Definition 1.1, (i)]. Write G for the underlying semi-graph of G, Π G for the [pro-Σ] fundamental group of G, and Π tp G for the tempered fundamental group of G [cf. [SemiAn], Example 2.10; the discussion preceding [SemiAn], Proposition 3.6]. Thus, we have a natural outer injection Π tp G → Π G — cf. [CbTpIII], Lemma 3.2, (i); the proof of [CbTpIII], Proposition 3.3, (i), (ii). Let us write G  −→ G, G  tp −→ G 14 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI for the universal pro-Σ and pro-tempered coverings of G corresponding to Π G , Π tp G and VCN( G  tp ) = lim VCN(H tp ) ←− tp — where H (respectively, H ) ranges over the subcoverings of G  → G (respectively, G  tp → G) corresponding to open subgroups of Π G (re- spectively, Π tp G ), and VCN(−) denotes the “VCN(−)” of the under- lying semi-graph of the semi-graph of anabelioids in parentheses [cf. Definition 1.1, (i), below; [NodNon], Definition 1.1, (iii)]. We begin by reviewing certain well-known facts concerning semi- graphs and group actions on semi-graphs.  = lim VCN(H), VCN( G) ←− def def Definition 1.1. Let Γ be a semi-graph [cf. the discussion at the be- ginning of [SemiAn], §1]. (i) We shall write Vert(Γ) (respectively, Cusp(Γ); Node(Γ)) for the set of vertices (respectively, open edges, i.e., “cusps”; closed def edges, i.e., “nodes”) of Γ. We shall write Edge(Γ) = Cusp(Γ) def Node(Γ); VCN(Γ) = Vert(Γ)  Edge(Γ). (ii) We shall write V Γ : Edge(Γ) −→ 2 Vert(Γ) (respectively, C Γ : Vert(Γ) −→ 2 Cusp(Γ) ; N Γ : Vert(Γ) −→ 2 Node(Γ) ; E Γ : Vert(Γ) −→ 2 Edge(Γ) ) [cf. (i); the discussion entitled “Sets” in [CbTpI], §0] for the map obtained by sending e ∈ Edge(Γ) (respectively, v ∈ Vert(Γ); v ∈ Vert(Γ); v ∈ Vert(Γ)) to the set of vertices (respectively, open edges; closed edges; edges) of Γ to which e abuts (respectively, which abut to v; which abut to v; which abut to v). For sim- plicity, we shall write V (resp C; N ; E) instead of V Γ (resp C Γ ; N Γ ; E Γ ) when there is no danger of confusion. (iii) Let n be a nonnegative integer; v, w ∈ Vert(Γ) [cf. (i)]. Then we shall write δ(v, w) ≤ n if the following conditions are satis- fied: • If n = 0, then v = w. • If n ≥ 1, then either δ(v, w) ≤ n−1 or there exist n closed edges e 1 , . . . , e n ∈ Node(Γ) of Γ [cf. (i)] and n + 1 vertices v 0 , . . . , v n ∈ Vert(Γ) of Γ such that v 0 = v, v n = w, and, for 1 ≤ i ≤ n, it holds that V(e i ) = {v i−1 , v i } [cf. (ii)]. Moreover, we shall write δ(v, w) = n if δ(v, w) ≤ n but δ(v, w) ≤ n − 1. If δ(v, w) = n, then we shall say that the distance be- tween v and w is equal to n. COMBINATORIAL ANABELIAN TOPICS IV 15 Definition 1.2. Let Γ be a semi-graph. (i) Let G be a group that acts on Γ. Then [by a slight abuse of notation, relative to the notation defined in the discussion entitled “Sets” in §0] we shall write Γ G for the semi-graph [i.e., the “G-invariant portion of Γ”] defined as follows: • We take Vert(Γ G ) to be Vert(Γ) G [cf. Definition 1.1, (i); the discussion entitled “Sets” in §0]. • We take Edge(Γ G ) to be Edge(Γ) G [cf. Definition 1.1, (i); the discussion entitled “Sets” in §0]. • Let e ∈ Edge(Γ G ) = Edge(Γ) G . Then the coincidence map ζ e : e −→ Vert(Γ G ) ∪ {Vert(Γ G )} of Γ G [cf. item (3) of the discussion at the beginning of [SemiAn], §1] is defined as follows: Write ζ e Γ : e → Vert(Γ) ∪ {Vert(Γ)} for the coincidence map associated to Γ. Then, for b ∈ e, if b ∈ e G and ζ e Γ (b) ∈ Vert(Γ) G (respectively, if either b ∈ e G or ζ e Γ (b) ∈ Vert(Γ) G ), then def def we set ζ e (b) = ζ e Γ (b) (respectively, = Vert(Γ G )). In par- ticular, it holds that V Γ G (e) = V Γ (e)∩Vert(Γ) G [cf. Defini- tion 1.1, (ii)] whenever it holds either that Γ is untangled [i.e., every node abuts to two distinct vertices — cf. the discussion entitled “Semi-graphs” in [NodNon], §0] or that G acts on Γ without inversion [i.e., that if e ∈ Edge(Γ) G , then e = e G ]. (ii) We shall write Γ ÷ for the semi-graph [i.e., the result of “subdividing” Γ] defined as follows: • We take Vert(Γ ÷ ) to be Vert(Γ)  Edge(Γ). • We take Edge(Γ ÷ ) to be the set of branches of Γ. • Let b be a branch of an edge e of Γ. Write e(b) ∈ Edge(Γ ÷ ), v(e) ∈ Vert(Γ ÷ ) for the edge and vertex of Γ ÷ correspond- ing to b, e, respectively. If b abuts, relative to Γ, to a ver- tex v ∈ Vert(Γ), then we take the edge e(b) to be a node that abuts to v(e) and the vertex of Γ ÷ corresponding to v ∈ Vert(Γ). If b does not abut, relative to Γ, to a vertex of Γ, then we take the edge e(b) to be a cusp that abuts to v(e). 16 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Definition 1.3. Let Γ be a semi-graph and Γ 0 ⊆ Γ a sub-semi-graph [cf. [SemiAn], the discussion following the figure entitled “A Typical Semi-graph”] of Γ. (i) We shall write Γ  0 ⊆ Γ for the sub-semi-graph of Γ [i.e., whenever a suitable condi- tion is satisfied [cf. Lemma 1.4, (v), below], a sort of “open neighborhood” of Γ 0 ] whose sets of vertices and edges are de- fined as follows. [Here, we recall that it follows immediately from the definition of a sub-semi-graph that a sub-semi-graph is completely determined by its sets of vertices and edges.] • We take Vert(Γ  0 ) to be Vert(Γ 0 ). • We take Edge(Γ  0 ) to be the set of edges e of Γ such that V Γ (e) ∩ Vert(Γ 0 )  = ∅. (ii) We shall write Γ ∈ 0 ⊆ Γ for the sub-semi-graph of Γ whose sets of vertices and edges are taken to be Vert(Γ)\Vert(Γ 0 ), Edge(Γ)\Edge(Γ 0 ), respectively. def (iii) We shall write Γ ∈ 0  = (Γ ∈ 0 )  [cf. (i), (ii)]. (iv) We shall say that an edge e of Γ is a Γ 0 -bridge if V Γ (e) ∩ Vert(Γ 0 ), V Γ (e) ∩ Vert(Γ ∈ 0 )  = ∅. [Thus, one verifies easily that every Γ 0 -bridge is a node.] We shall write Brdg(Γ 0 ⊆ Γ) ⊆ Node(Γ) for the set of Γ 0 -bridges of Γ. By abuse of notation, we shall write Brdg(Γ 0 ⊆ Γ) ⊆ Γ for the sub-semi-graph of Γ whose sets of vertices and edges are taken to be ∅ [i.e., the empty set], Brdg(Γ 0 ⊆ Γ) ⊆ Node(Γ), respectively. Lemma 1.4 (Basic properties of sub-semi-graphs). Let Γ be a semi-graph, Γ 0 ⊆ Γ a sub-semi-graph [cf. [SemiAn], the discussion fol- lowing the figure entitled “A Typical Semi-graph”] of Γ, G a group, and ρ : G → Aut(Γ) an action of G on Γ. Then the following hold: (i) Suppose either that Γ is untangled or that G acts on Γ with- out inversion. Then the semi-graph Γ G [cf. Definition 1.2, (i)] may be regarded, in a natural way, as a sub-semi-graph of Γ. (ii) Suppose that G acts on Γ without inversion, and that every edge of Γ abuts to at least one vertex of Γ. Then every edge of Γ G abuts to at least one vertex of Γ G . (iii) The semi-graph Γ ÷ [cf. Definition 1.2, (ii)] is untangled. (iv) There exists a natural injection Aut(Γ) → Aut(Γ ÷ ). More- over, the resulting action ρ ÷ of G on Γ ÷ [i.e., the composite ρ G → Aut(Γ) → Aut(Γ ÷ )] is an action without inversion. Finally, it holds that Γ G = ∅ if and only if (Γ ÷ ) G = ∅. COMBINATORIAL ANABELIAN TOPICS IV 17 (v) Suppose that every edge of Γ 0 abuts to at least one vertex of Γ 0 . Then Γ 0 may be regarded, in a natural way, as a sub- semi-graph of Γ  0 [cf. Definition 1.3, (i)]. (vi) We have an equality of subsets of Edge(Γ): ∈  Edge(Γ  0 ) ∩ Edge(Γ 0 ) = Brdg(Γ 0 ⊆ Γ). Proof. The assertions of Lemma 1.4 follow immediately from the vari- ous definitions involved.  Lemma 1.5 (Sub-semi-graphs of invariants). In the situation of Lemma 1.4, suppose either that Γ is untangled or that G acts on Γ without inversion. Suppose, moreover, that the sub-semi-graph Γ 0 ⊆ Γ is a connected component of the sub-semi-graph Γ G ⊆ Γ [cf. Lemma 1.4, (i)]. Then the following hold: (i) The action ρ naturally determines actions of G on Γ 0 , Γ  0 , ∈  Γ 0 , respectively. (ii) The intersection of Γ  0 ⊆ Γ with any connected component of Γ G ⊆ Γ that is  = Γ 0 is empty. (iii) We have an equality of subsets of Edge(Γ): Edge(Γ G ) ∩ Brdg(Γ  0 ⊆ Γ) = ∅. Proof. The assertions of Lemma 1.5 follow immediately from the vari- ous definitions involved.  Lemma 1.6 (Existence of fixed points). Let Γ be a finite con- nected [hence nonempty] semi-graph, G a finite solvable group whose order is a Σ-integer [cf. the discussion entitled “Numbers” in §0], and ρ : G −→ Aut(Γ) for the [discrete] topological funda- an action of G on Γ. Write Π disc Γ Σ  disc → Γ, mental group of Γ; Π Γ for the pro-Σ completion of Π disc Γ ; Γ  Σ → Γ for the discrete, pro-Σ universal coverings of Γ corresponding Γ Σ   to Π disc Γ , Π Γ , respectively. Let  ∈ {disc, Σ}. Write Aut( Γ → Γ) ⊆   such that α   ) for the group of automorphisms α  of Γ  lies over Aut( Γ a(n) [necessarily unique] automorphism α of Γ;   → Γ) −→ Aut(Γ) Aut( Γ α  → α for the resulting natural homomorphism;   Π  Γ//G = Aut( Γ → Γ) × Aut(Γ) G def 18 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI   → Γ) → for the fiber product of the natural homomorphism Aut( Γ Aut(Γ) and the action ρ : G → Aut(Γ). Thus, one verifies easily that Π  Γ//G fits into an exact sequence  1 −→ Π  Γ −→ Π Γ//G −→ G −→ 1. Let s : G → Π  Γ//G be a section of the above exact sequence. Write    ) for the action obtained by forming the compos- ρ  s : G → Aut( Γ pr 1 s   → Γ) → Aut( Γ   ). We shall say that a ite G → Π  → Aut( Γ Γ//G  Σ → Γ is G-compatible if connected finite subcovering Γ ∗ → Γ of Γ Γ ∗ → Γ is Galois, and, moreover, the corresponding normal open sub- Σ group of Π Σ Γ is preserved by the outer action of G, via ρ, on Π Γ . If  Σ → Γ, then Γ ∗ → Γ is a G-compatible connected finite subcovering of Γ let us write ρ s,∗ : G → Aut(Γ ∗ ) for the action of G on Γ ∗ determined by G ρ   s ; Γ ∗ for the semi-graph defined in Definition 1.2, (i), with respect to the action ρ s,∗ . [Thus, if Γ, hence also Γ ∗ , is untangled, then Γ G ∗ is a sub-semi-graph of Γ ∗ — cf. Lemma 1.4, (i).] Then the following hold: (i) Suppose that Γ is untangled. Then, for each G-compatible  Σ → Γ, the sub- connected finite subcovering Γ ∗ → Γ of Γ G semi-graph Γ ∗ ⊆ Γ ∗ coincides with the disjoint union of some def [possibly empty] collection of connected components of Γ ∗ | Γ G = Γ ∗ × Γ Γ G ⊆ Γ ∗ . (ii) Suppose that Γ is untangled, and that G is isomorphic to Z/lZ for some prime number l ∈ Σ. Then, for every G-  Σ → Γ, compatible connected finite subcovering Γ ∗ → Γ of Γ the sub-semi-graph Γ G ∗ ⊆ Γ ∗ is nonempty.  disc ) G for the sub-semi-graph (iii) Suppose that  = disc. Write ( Γ [cf. Lemma 1.4, (i)] of [the necessarily untangled semi-graph!]  disc defined in Definition 1.2, (i), with respect to the action Γ  disc ) G is nonempty and connected. If, more- ρ  disc s . Then ( Γ over, we write (Γ G ) 0 ⊆ Γ G for the image of the composite  disc → Γ, then the resulting morphism ( Γ  disc ) G →  disc ) G → Γ ( Γ (Γ G ) 0 is a [discrete] universal covering of (Γ G ) 0 . (iv) Suppose that  = disc (respectively,  = Σ). Then the set  disc ) G VCN( Γ  Σ ) G = lim VCN(Γ ∗ ) G ) (respectively, VCN( Γ ←− def — where, in the resp’d case, the projective limit is taken over  Σ → the G-compatible connected finite subcoverings Γ ∗ → Γ of Γ Γ — is nonempty. (v) Suppose that  = Σ, that Γ is untangled, and that G is isomorphic to Z/lZ for some prime number l ∈ Σ. Let (Γ G ) 0 ⊆ COMBINATORIAL ANABELIAN TOPICS IV 19 Γ G be a [nonempty] connected component of Γ G such that    Σ ) G → VCN(Γ)  = ∅ VCN((Γ G ) 0 ) ∩ Im VCN( Γ [cf. (iv)]. Then there exists a G-compatible connected finite  Σ → Γ such that the image of Γ G subcovering Γ ∗ → Γ of Γ ∗ ⊆ Γ ∗ G G in Γ coincides with (Γ ) 0 ⊆ Γ . (vi) Suppose that  = Σ, and that Γ is untangled. Then the  Σ determined by the projective  Σ ) G of Γ sub-pro-semi-graph ( Γ system of sub-semi-graphs Γ G ∗ — where Γ ∗ → Γ ranges over  Σ → Γ — is the G-compatible connected finite subcoverings of Γ nonempty and connected. If, moreover, we write (Γ G ) 0 ⊆  Σ ) G → Γ  Σ → Γ, then Γ G for the image of the composite ( Γ  Σ ) G → (Γ G ) 0 is a pro-Σ universal the resulting morphism ( Γ G covering of (Γ ) 0 . Proof. First, we verify assertion (i). Let us first observe that one verifies immediately that there is an inclusion of sub-semi-graphs Γ G ∗ ⊆ Γ ∗ | Γ G [cf. Lemma 1.4, (i)]. Next, let us observe that it follows immediately from Lemma 1.4, (iii), (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of generality that G acts without inversion on Γ [which implies that G acts trivially on Γ G — cf. Definition 1.2, (i)]. Thus, to complete the verification of assertion (i), it suffices to verify that the following assertion holds: Claim 1.6.A: Let (Γ ∗ | Γ G ) 0 ⊆ Γ ∗ | Γ G be a connected com- ponent of Γ ∗ | Γ G such that (Γ ∗ | Γ G ) 0 ∩ Γ ∗ G  = ∅. Then (Γ ∗ | Γ G ) 0 ⊆ Γ G ∗ . To verify Claim 1.6.A, let us observe that since (Γ ∗ | Γ G ) 0 ∩ Γ G ∗  = ∅, the action ρ s,∗ of G on Γ ∗ stabilizes (Γ ∗ | Γ G ) 0 ⊆ Γ ∗ . In particular, we obtain an action of G on (Γ ∗ | Γ G ) 0 over Γ G . Thus, since the action of G on Γ G is trivial, and the composite (Γ ∗ | Γ G ) 0 → Γ ∗ | Γ G → Γ G is a connected finite covering of some connected component of Γ G , again by our assumption that (Γ ∗ | Γ G ) 0 ∩ Γ G ∗  = ∅, we conclude that the action of G on (Γ ∗ | Γ G ) 0 is trivial, i.e., that there is an inclusion of sub-semi-graphs (Γ ∗ | Γ G ) 0 ⊆ Γ G ∗ . This completes the proof of Claim 1.6.A, hence also of assertion (i). Next, we verify assertion (ii). One verifies immediately that we may assume without loss of generality that Γ ∗ = Γ. Now suppose that Γ G = ∅. Then since G ∼ = Z/lZ, it follows that the action of G on Γ is free, which thus implies that the quotient map Γ  Γ/G is a covering of Γ/G. In particular, Π Σ Γ//G is isomorphic to the pro-Σ completion of the topological fundamental group of the semi-graph Γ/G. Thus, the pro-Σ group Π Σ Γ//G is free, hence, in particular, torsion-free. But this contradicts the existence of the section of the surjection Π Σ Γ//G  G determined by s. This completes the proof of assertion (ii). 20 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Next, we verify the resp’d portion of assertion (iv) [i.e., the assertion  Σ ) G  = ∅] in the case where G is isomorphic to Z/lZ for some that VCN( Γ prime number l ∈ Σ. Let us first observe that it follows immediately from Lemma 1.4, (iii), (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of generality that Γ is untangled. Thus, the assertion that  Σ ) G  = ∅ follows immediately from assertion (ii), together with VCN( Γ the well-known elementary fact that a projective limit of nonempty finite sets is nonempty. This completes the proof of the assertion that  Σ ) G  = ∅ in the case where G is isomorphic to Z/lZ for some VCN( Γ prime number l ∈ Σ.  disc Next, we verify assertion (iii). Let us first observe that since Γ  disc ) G is a tree, hence untangled, it follows from Lemma 1.4, (i), that ( Γ  disc . Next, let us observe that it follows im- is a sub-semi-graph of Γ mediately from Lemma 1.4, (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of generality that G acts without inversion on Γ.  disc ) G is nonempty and connected follows im- Thus, the assertion that ( Γ mediately from [SemiAn], Lemma 1.8, (ii). The remainder of assertion (iii) follows from a similar argument to the argument applied in the proof of assertion (i). This completes the proof of assertion (iii). In particular, the unresp’d portion of assertion (iv) [i.e., the assertion that  disc ) G  = ∅] holds. VCN( Γ Next, we verify assertion (v). Let us first observe that, to verify assertion (v), it follows immediately from Lemma 1.4, (iii), (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of generality that the action ρ is an action without inversion, and that every edge of Γ abuts to at least one vertex of Γ. In particular, since [we have assumed that] (Γ G ) 0  = ∅, it follows from Lemma 1.4, (ii), (v), that (Γ G )  0  = ∅ [cf. Definition 1.3, (i)]. Now if Γ G is connected, then one verifies imme- id diately that the trivial covering Γ → Γ satisfies the condition imposed on “Γ ∗ → Γ” in the statement of assertion (v). Thus, to complete the verification of assertion (v), we may assume without loss of generality that Γ G is not connected, hence [cf. Lemma 1.4, (ii)] contains at least one vertex ∈ Vert((Γ G ) 0 ). In particular, (Γ G ) ∈ 0   = ∅ [cf. Definition 1.3, (iii)].  Write ((Γ G )  ) → (Γ G )  0 0 for the trivial Z/lZ-covering obtained by taking a disjoint  union of copies of (Γ G )  0 indexed by the elements of G ∈  G ∈  Z/lZ; ((Γ ) 0 ) → (Γ ) 0 for the trivial Z/lZ-covering obtained by taking a disjoint union of copies of (Γ G ) ∈ 0  indexed  by the elements  G ∈  [cf. of Z/lZ. Then the natural actions of G on ((Γ G )  0 ) , ((Γ ) 0 )  G  Lemma 1.5, (i)] determine natural actions of G × Z/lZ on ((Γ ) ) , 0  G ∈  ((Γ ) 0 ) , i.e., we have homomorphisms    , ) ρ  : G × Z/lZ −→ Aut ((Γ G )  0 COMBINATORIAL ANABELIAN TOPICS IV 21    ρ ∈  : G × Z/lZ −→ Aut ((Γ G ) ∈ 0  ) . ∼ Let φ : G → Z/lZ be an isomorphism. Write ρ ∈ φ  :    ρ ∈  G × Z/lZ −→ G × Z/lZ −→ Aut ((Γ G ) ∈ 0  ) (a, b) → (a, φ(a) + b) for the composite of ρ ∈  with the homomorphism described in the second line of the display. def Next, for e ∈ Brdg = Brdg((Γ G ) 0 ⊆ Γ) [cf. Definition 1.3, (iv)], write G · e ⊆ Edge((Γ G )  0 ) for the G-orbit of e. Then it is immediate that G · e ⊆ Brdg; moreover, since G ∼ = Z/lZ, it follows immediately from Lemma 1.5, (iii), that G · e is a G-torsor. Next, let us write  def  def  G  ((Γ G )  G · e, 0 ) | G·e = ((Γ ) 0 ) × (Γ G )  0  ((Γ G ) ∈ 0  ) | G·e = ((Γ G ) ∈ 0  ) × (Γ G ) ∈  G · e 0 [cf. Lemma 1.4, (vi)]. Then one verifies easily from the various defini- tions involved that the following hold:   ) , ((Γ G ) ∈ 0  ) (a) The actions ρ  , ρ ∈ φ  of G × Z/lZ on ((Γ G )  0    G ∈  determine actions on  these fibers ((Γ G ) ) | , ((Γ ) ) | G·e . G·e 0  0 G  G ∈  (b) These fibers ((Γ ) 0 ) | G·e , ((Γ ) 0 ) | G·e are (G×Z/lZ)-torsors with respect to the actions of (a).  ∼ (c) There is a  natural isomorphism of semi-graphs ((Γ G )  ) | G·e → 0 ((Γ G ) ∈ 0  ) | G·e [cf. Lemma 1.4, (vi)], which we shall use to iden- tify these two semi-graphs.   G ∈  (d) Let e base ∈ ((Γ G )  ) | = ((Γ ) ) | G·e [cf. (c)] be a lifting G·e 0 0 of e ∈ Brdg. Then there is a uniquely determined [cf. (b)] isomorphism  ∼  G ∈  ι e base : ((Γ G )  0 ) | G·e −→ ((Γ ) 0 ) | G·e of (G × Z/lZ)-torsors [cf. (b)] that maps e base to e base . Let B be a collection of elements “e base ” as in (d) such that the map e base → e determines a bijection between B and  the set of G-orbits  G ∈  of Brdg. Thus, by gluing ((Γ G )  ) to ((Γ ) ) by means of the 0 0 collection of isomorphisms {ι e base } e base ∈B of (d) [cf. Lemma 1.4, (vi)], we obtain a finite covering Γ ∗ → Γ, together with an action of G×Z/lZ on Γ ∗ [i.e., obtained by gluing the actions ρ  , ρ ∈ φ  ], such that the morphism Γ ∗ → Γ is equivariant with respect to this action of G×Z/lZ on Γ ∗ and the action of G × Z/lZ on Γ obtained by composing the projection G × Z/lZ → G with the given action of G on Γ. Next, let us observe that since φ is an isomorphism, and both (Γ G ) 0 and (Γ G ) ∈ 0  contain vertices fixed by G [cf. the discussion at the beginning of the present proof of assertion (v)], one verifies immediately — e.g., by considering the orbit by the action of G × {1} (⊆ G × Z/lZ) of some 22 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI lifting to Γ ∗ [which may be chosen to pass through an element of B] of a path of minimal length between such vertices fixed by G — that Γ ∗ is connected. Moreover, it follows from the definition of Γ ∗ that the covering Γ ∗ → Γ is Galois, G-compatible, and equipped with a natural ∼  Σ → Γ factors as a isomorphism Gal(Γ ∗ /Γ) → Z/lZ; in particular, Γ  Σ → Γ ∗ → Γ. composite Γ Next, let us observe that, for each g ∈ G, the automorphism α g of Γ ∗ obtained by considering the difference between ρ s,∗ (g) and the action of g [i.e., (g, 0) ∈ G × Z/lZ] on Γ ∗ defined above is an automorphism over Γ. Moreover, it follows immediately from our assumption that    Σ ) G → VCN(Γ)  = ∅ VCN((Γ G ) 0 ) ∩ Im VCN( Γ that α g fixes an element of VCN(Γ ∗ ) that maps to VCN((Γ G ) 0 ) ⊆ VCN(Γ). But this implies that α g is trivial, i.e., that the action ρ s,∗ of G coincides with the action of G (= G × {0} ⊆ G × Z/lZ) on Γ ∗ defined above. On the other hand, since φ is an isomorphism, it follows that (Γ ∗ ) G  ⊆ Γ ∗ is contained in the sub-semi-graph of Γ ∗ determined by ((Γ G )  0 ) . In particular, it follows immediately from Lemma 1.5, (ii), that the G G image of Γ G ∗ ⊆ Γ ∗ in Γ is contained in (Γ ) 0 ⊆ Γ . Thus, it follows immediately from assertion (i) that the image of Γ G ∗ ⊆ Γ ∗ in Γ coincides G G with (Γ ) 0 ⊆ Γ . This completes the proof of assertion (v). Next, we verify assertion (vi). First, we claim that the following assertion holds: Claim 1.6.B: If G is isomorphic to Z/lZ for some prime number l ∈ Σ, then assertion (vi) holds. Indeed, it follows from the resp’d portion of assertion (iv) [i.e., the  Σ ) G  = ∅] in the case where G is isomorphic to assertion that VCN( Γ Z/lZ for some prime number l ∈ Σ [i.e., the case that has already  Σ ) G  = ∅. On the other hand, it follows im- been verified!] that ( Γ mediately from assertion (v) [i.e., by allowing “Γ” to vary among the  Σ → Γ] that ( Γ  Σ ) G is G-compatible connected finite subcoverings of Γ connected. Thus, the final portion of assertion (vi) [in the case where G is isomorphic to Z/lZ for some prime number l ∈ Σ] follows imme- diately from assertion (i) [and the evident pro-Σ version of [SemiAn], Proposition 2.5, (i)]. This completes the proof of Claim 1.6.B. Next, we verify assertion (vi) for arbitrary finite solvable G by in- duction on G  . Let us first observe that it follows immediately from Lemma 1.4, (iii), (iv), that, by replacing Γ by Γ ÷ , we may assume with- out loss of generality that the action ρ is an action without inversion. Next, observe that since G is finite and solvable, there exists a normal subgroup N ⊆ G of G such that G/N is a nontrivial finite group of prime order. Then it follows from the induction hypothesis that if we COMBINATORIAL ANABELIAN TOPICS IV 23 write (Γ N ) 0 ⊆ Γ N for the [nonempty, connected!] image of the com-  Σ → Γ, then the resulting morphism ( Γ  Σ ) N → (Γ N ) 0  Σ ) N → Γ posite ( Γ is a pro-Σ universal covering of (Γ N ) 0 , and, moreover, [since the action  Σ ) N . Next, let ρ is an action without inversion] N acts trivially on ( Γ us observe that since N is normal in G, [one verifies immediately that]  Σ  Σ N ⊆ Γ  Σ . Thus, by replacing the action ρ  Σ s of G on Γ preserves ( Γ )  Σ ) N → (Γ N ) 0 , G/N ) and applying Claim 1.6.B, we  Σ → Γ, G) by (( Γ ( Γ conclude that assertion (vi) holds for the given G. This completes the proof of assertion (vi). Finally, we verify the resp’d portion of assertion (iv) [i.e., the as-  Σ ) G  = ∅]. Let us first observe that, to verify the sertion that VCN( Γ  Σ ) G  = ∅, it follows immediately from Lemma 1.4, assertion that VCN( Γ (iii), (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of  Σ ) G  = ∅ generality that Γ is untangled. Thus, the assertion that VCN( Γ follows immediately from assertion (vi). This completes the proof of Lemma 1.6.  Remark 1.6.1. The conclusion of Lemma 1.6, (vi), follows for an arbitrary [i.e., not necessarily solvable!] finite group G from [ZM], The- orems 2.8, 2.10. That is to say, the proof given above of Lemma 1.6, (vi), may be regarded as an alternative proof of these results of [ZM] in the case where G is solvable. In this context, it is also perhaps of inter- est to observe that, by considering Lemma 1.6, (vi), in the case where Σ = Primes and “Γ” is taken to be some finite connected sub-semi-  disc that is stabilized by the action of G [where we note that graph of Γ  disc is a union of such sub-semi-graphs], one one verifies easily that Γ obtains an alternative proof of the classical result concerning actions of finite groups on trees quoted in the proofs of Lemma 1.6, (iii); [SemiAn], Lemma 1.8, (ii) — hence also alternative proofs of Lemma 1.6, (iii); [SemiAn], Lemma 1.8, (ii) — in the case where the finite group under consideration is solvable. Remark 1.6.2. (i) In the situation of Lemma 1.6, if G is isomorphic to Z/l n Z for some prime number l ∈ Σ and a positive integer n, then the conclusion of the resp’d portion of Lemma 1.6, (iv), may be verified by the following easier argument: Since [as is well- known] a projective limit of nonempty finite sets is nonempty,  Σ ) G  = ∅, it suffices to verify to verify the assertion that VCN( Γ that VCN(Γ ∗ ) G  = ∅ for every G-compatible connected finite  Σ → Γ. Moreover, one verifies im- subcovering Γ ∗ → Γ of Γ mediately that we may assume without loss of generality that 24 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Γ ∗ = Γ. Next, let us observe that it follows immediately from Lemma 1.4, (iv), that, by replacing Γ by Γ ÷ , we may assume without loss of generality that G acts on Γ without inversion. def Write H ⊆ G for the unique subgroup such that Q = G/H def is of order l; Γ Q = Γ/H for the “quotient semi-graph”, i.e., the semi-graph whose vertices, edges, and branches are, re- spectively, the H-orbits of the vertices, edges, and branches of Γ [cf. the fact that G acts on Γ without inversion]. Then one verifies immediately that the natural morphism of semi-graphs Γ  Γ Q determines an outer homomorphism Σ Π Σ Γ//G −→ Π Γ Q //Q [cf. the notation of the statement of Lemma 1.6]. Now since Π Σ Γ Q is a free pro-Σ group, hence torsion-free, it follows that the restriction s(H) → Π Σ Γ Q //Q [which clearly factors through Σ Σ Σ Π Γ Q ⊆ Π Γ Q //Q ] of the outer homomorphism Π Σ Γ//G → Π Γ Q //Q to s(H) ⊆ Π Σ Γ//G is trivial, hence that s determines a section Σ s Q : Q → Π Γ Q //Q of the natural surjection Π Σ Γ Q //Q  Q. In particular, by applying Lemma 1.6, (ii), we thus conclude that VCN(Γ Q ) Q  = ∅. Let z Q ∈ VCN(Γ Q ) Q , z ∈ VCN(Γ) a lifting of z Q , and g ∈ G a generator of G. Then since Q fixes z Q , it follows that z g = z h , for some h ∈ H, hence that z is fixed by g · h −1 ∈ G. On the other hand, since g · h −1 generates G, we thus conclude that z is fixed by G, i.e., that VCN(Γ ∗ ) G  = ∅, as desired. (ii) The proof of Lemma 1.6, (ii), as well as the argument of (i) above, is essentially the same as the argument applied in [AbsCsp] to prove [AbsCsp], Lemma 2.1, (iii). Remark 1.6.3. In the respective situations of Lemma 1.6, (iii), (vi),  Σ ) G are nec-  disc ) G and the sub-pro-semi-graph ( Γ the sub-semi-graph ( Γ essarily connected [cf. Lemma 1.6, (iii), (vi)]. On the other hand, Γ G is not, in general, connected. This phenomenon may be seen in the follow-  disc is the graph given by ing example: Suppose that 2 ∈ Σ, and that Γ the integral points of the real line R, i.e., the vertices are given by the elements of Z ⊆ R, and the edges are given by the closed line segments joining adjacent elements of Z. For N = 2M a positive even integer,  disc by the evident action of N ∈ Z on write Γ N for the quotient of Γ  disc via translations. Thus, we have a diagram of natural covering Γ maps  disc −→ Γ N −→ Γ def Γ = Γ 2 , COMBINATORIAL ANABELIAN TOPICS IV 25 and the group G = Z/2Z acts equivariantly on this diagram via mul- tiplication by ±1. Here, we observe that since N is even, one verifies immediately that G acts on Γ N without inversion. Then one computes easily that  disc ) G = {0}, Γ G = M Z/N Z. ( Γ N Σ G  In particular, the pro-semi-graph ( Γ ) corresponds to the inverse limit lim M Z/N Z, ←− hence consists of a single pro-vertex and no pro-edge and, in particular, is nonempty and connected. On the other hand, each Γ G N consists of precisely two vertices and no edges, hence is not connected. Definition 1.7. Let G be a profinite group and ρ : G → Aut(G) a continuous homomorphism. (i) We shall say that ρ is of ENN-type [where the “ENN” stands for “extended NN”] (respectively, of EPIPSC-type [where the “EPIPSC” stands for “extended PIPSC”]) if there exists a nor- mal closed subgroup I G ⊆ G of G such that, for every open ρ subgroup J ⊆ I G of I G , the composite J → G → Aut(G) factors as a composite J  J Σ-ab-free → Aut(G) [cf. the dis- cussion entitled “Groups” in §0], where the second arrow is of NN-type [cf. [NodNon], Definition 2.4, (iii)] (respectively, of PIPSC-type [cf. [CbTpIII], Definition 1.3]). In this situation, we shall refer to I G as a conducting subgroup. Suppose that ρ is of ENN-type for some conducting subgroup I G ⊆ G. Then we shall say that ρ is verticially quasi-split if there exists an open subgroup H ⊆ G that acts as the identity [i.e., relative to the action induced by ρ] on the underlying semi-graph G of G and, moreover, for every v ∈ Vert(G), satisfies the following condition: the extension of profinite groups [cf. the discussion entitled “Topological groups” in [CbTpI], §0] out 1 −→ Π v −→ Π v  H −→ H −→ 1 — where Π v ⊆ Π G is a verticial subgroup associated to v ∈ Vert(G) — associated to the outer action of H on Π v de- termined by ρ [cf. [CmbGC], Proposition 1.2, (ii); [CbTpI], out Lemma 2.12] admits a splitting s v : H → Π v  H such that the image of the restriction of s v to I G ∩ H commutes with Π v . (ii) Let l ∈ Σ. Then we shall say that ρ is l-cyclotomically full if χ G ρ  Σ ) ×  Z × [cf. the image of the composite G → Aut(G) → ( Z l [CbTpI], Definition 3.8, (ii)] is open. 26 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Remark 1.7.1. It follows immediately from [CbTpIII], Remark 1.6.2, that the following implication holds: EPIPSC-type =⇒ ENN-type. Lemma 1.8 (Outer representations induced on pro-l comple- tions). Let G be a profinite group and ρ : G → Aut(G) a continu- ous homomorphism of ENN-type (respectively, of EPIPSC-type) for a conducting subgroup I G ⊆ G [cf. Definition 1.7, (i)]. For l ∈ Σ, write G {l} for the semi-graph of anabelioids of pro-{l} PSC-type obtained by forming the pro-l completion of G [cf. [SemiAn], Defini- ρ tion 2.9, (ii)]. Then the composite G → Aut(G) → Aut(G {l} ) is of ENN-type (respectively, of EPIPSC-type) for some conducting subgroup ⊆ G, which may be taken to be a normal open subgroup of I G . Proof. This follows immediately from the various definitions involved [cf. also [CbTpI], Theorem 4.8, (iv); [CbTpI], Corollary 5.9, (ii), (iii)].  Definition 1.9. Let z ∈ VCN(G). If z ∈ Vert(G) (respectively, z ∈ Edge(G)), then we shall refer to a verticial (respectively, an edge-like) subgroup of Π tp G associated to z [cf. [SemiAn], Theorem 3.7, (i), (iii)] as a VCN-subgroup of Π tp  ∈ VCN( G  tp ), we shall G associated to z. For z tp also speak of VCN-subgroups of Π G associated to z  . Definition 1.10. (i) Let Γ be a semi-graph and v ∈ Vert(Γ). Then we shall write V δ≤1 (v) ⊆ Vert(Γ) for the subset consisting of w ∈ Vert(Γ) such that either w = v or N (v) ∩ N (w)  = ∅. Also, we shall def write Star(v) = V δ≤1 (v)  E(v) ⊆ VCN(Γ). (ii) Let v ∈ Vert(G). Then we shall write V δ≤1 (v) ⊆ Vert(G), Star(v) ⊆ VCN(G) for the respective subsets of (i) applied to the underlying semi-graph of G.  Then we shall write V δ≤1 (  (iii) Let v  ∈ Vert( G). v ) ⊆ Vert( G),  for the respective projective limits of the Star( v ) ⊆ VCN( G) subsets of (ii), i.e., where the constructions of these subsets are applied to the images of v  in the connected finite etale subcoverings of G  → G. COMBINATORIAL ANABELIAN TOPICS IV 27 Lemma 1.11 (VCN-subgroups and sections). Let G be a profinite  group, ρ : G → Aut(G) a continuous homomorphism, z  ∈ VCN( G), z  tp ∈ VCN( G  tp ), Π z  ⊆ Π G a VCN-subgroup of Π G associated to z  ∈  and Π z  tp ⊆ Π tp a VCN-subgroup of Π tp associated to z  tp VCN( G), G G def out out def tp [cf. Definition 1.9]. Write Π G = Π G  G, Π tp G = Π G  G [cf. the discussion entitled “Topological groups” in [CbTpI], §0]. Thus, we have a natural commutative diagram 1 −−−→ Π tp −−−→ Π tp −−−→ G −−−→ 1 ⏐ G ⏐ G  ⏐ ⏐     1 −−−→ Π G −−−→ Π G −−−→ G −−−→ 1 — where the horizontal sequences are exact; the vertical arrows are  tp  outer injections; Π tp G acts naturally on G ; Π G acts naturally on G. Then the following hold: (i) It holds that Π z  = N Π G (Π z  ) ∩ Π G = C Π G (Π z  ) ∩ Π G , def D z  = N Π G (Π z  ) = C Π G (Π z  ) = N Π G (D z  ) = C Π G (D z  ), tp Π z  tp = N Π tp (Π z  tp ) ∩ Π tp G = C Π tp (Π z  tp ) ∩ Π G , G G def D z  tp = N Π tp (Π z  tp ) = C Π tp (Π z  tp ) = N Π tp (D z  tp ) = C Π tp (D z  tp ). G G G G (ii) Suppose that ρ is of ENN-type for a conducting subgroup I G ⊆ G [cf. Definition 1.7, (i)]. Let S be a nonempty subset  and s : G → Π G a section of the surjection Π G  of VCN( G) G such that, for each y  ∈ S, it holds that s(I G ) ≺ D y  [cf. the discussion entitled “Groups” in §0]. Then there exists an  such that S ⊆ Star( element v  ∈ Vert( G) v ) [cf. Definition 1.10, (iii)]. (iii) Suppose that ρ is of ENN-type for a conducting subgroup I G ⊆ G. Let s : G → Π G be a section of the surjection Π G  G such that s(I G ) ≺ D z  [cf. the discussion entitled ”Groups” in def §0]. Write G s = C Π G (s(I G )). Then there exists an element  such that s(G) ⊆ G s ⊆ D z   . z   ∈ VCN( G) (iv) Suppose that ρ is of ENN-type for a conducting subgroup tp I G ⊆ G. Let s : G → Π tp G be a section of the surjection Π G  G such that s(I G ) ≺ D z  tp [cf. the discussion entitled ”Groups” in def §0]. Write G s = C Π tp (s(I G )). Then there exists an element G ( z  ) tp ∈ VCN( G  tp ) such that s(G) ⊆ G s ⊆ D ( z  ) tp . In par- ticular, G s is contained in a profinite subgroup of Π tp G [cf. (i)]. 28 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. First, we verify assertion (i). The two equalities of the first (respectively, third) line of the display and the first “=” of the sec- ond (respectively, fourth) line of the display follow immediately from [CmbGC], Proposition 1.2, (i), (ii) (respectively, [CmbGC], Proposi- tion 1.2, (i), (ii), together with the injection reviewed at the beginning of the present §1). Thus, the second and third “=” of the second (respectively, fourth) line of the display follow immediately from the chain of inclusions D z  ⊆ N Π G (D z  ) ⊆ C Π G (D z  ) ⊆ C Π G (D z  ∩ Π G ) = C Π G (Π z  ) = D z  (respectively, D z tp ⊆ N Π tp (D z  tp ) ⊆ C Π tp (D z  tp ) ⊆ C Π tp (D z  tp ∩Π tp G ) = C Π tp (Π z  tp ) = D z  tp ) G G G G — where the third “⊆” follows immediately from [CbTpII], Lemma 3.9, (i) (respectively, the [easily verified] tempered version of [CbTpII], Lemma 3.9, (i)). This completes the proof of assertion (i). Next, we verify assertion (ii). Let us first observe that it follows from the definition of the term “ENN-type” that the restriction of ρ to I G ⊆ G factors through the quotient I G  I G Σ-ab-free [cf. the discussion def out def entitled “Groups” in §0]. Write Π I G = Π G  I G and Π I G Σ-ab-free = out Π G  I G Σ-ab-free . Thus, we have a commutative diagram 1 −−−→ Π G −−−→    Π G ⏐ ⏐ −−−→ G ⏐ ⏐ −−−→ 1 1 −−−→ Π G −−−→    Π I G ⏐ ⏐  −−−→ I G ⏐ ⏐  −−−→ 1 1 −−−→ Π G −−−→ Π I G Σ-ab-free −−−→ I G Σ-ab-free −−−→ 1 — where the horizontal sequences are exact, the upper vertical arrows are injective, the lower vertical arrows are surjective, and the two right- hand squares are cartesian. Next, let us observe that we may assume  without loss of generality that S is equal to the set of all y  ∈ VCN( G) such that s(I G ) ≺ D y  . Now since s(I G ) ≺ D y  = C Π G (Π y  ) [cf. assertion (i)] for every y  ∈ S, it holds that, for each y  ∈ S, some open subgroup s of the image J ⊆ Π I G Σ-ab-free of I G → Π I G  Π I G Σ-ab-free is contained in C Π I Σ-ab-free (Π y  ). In particular, it follows from [NodNon], Propositions G 3.8, (i); 3.9, (i), (ii), (iii), that • every pair of edges ∈ S abut to a common vertex ∈ S; • the distance between any two vertices ∈ S is ≤ 2 [cf. Defini- tion 1.1, (iii)], and the edges “e 1 , . . . , e n ” and vertices “v 0 , . . . , v n ” of loc. cit. may be taken to be ∈ S; • if e  ∈ S is an edge, then V( e ) ⊆ S. COMBINATORIAL ANABELIAN TOPICS IV 29 It is now a matter of elementary combinatorial graph theory [cf. also [NodNon], Lemma 1.8] to conclude that S ⊆ Star( v ) for some v  ∈  Vert( G), as desired. This completes the proof of assertion (ii). Next, we verify assertion (iii). Since s(I G ) ≺ D z  , the action of  some open subgroup of I G on G  determined by s| I G fixes z  ∈ VCN( G). Thus, it follows from the definition of G s that, if, for γ ∈ G s , we write  for the image of z  by the action of γ ∈ G s , then the action z  γ ∈ VCN( G)  i.e., s(I G ) ≺ D z  γ of some open subgroup of I G on G  fixes z  γ ∈ VCN( G), for every γ ∈ G s .  Then it follows from assertion (ii) Now suppose that z  ∈ Edge( G).  such that { z  γ | γ ∈ G s } ⊆ E( v ). that there exists a vertex v  ∈ Vert( G) γ  Now if { z  | γ ∈ G s } = 1, then it is immediate that G s ⊆ D z  . On the other hand, if { z  γ | γ ∈ G s }  ≥ 2, then one verifies immediately from the various definitions involved [cf. also [NodNon], Lemma 1.8] that the  which thus implies that G s ⊆ D v  . This action of G s fixes v  ∈ Vert( G),  completes the proof of assertion (iii) in the case where z  ∈ Edge( G).  Then it follows from assertion (ii) Next, suppose that z  ∈ Vert( G).  such that that the set S δ of vertices v  ∈ Vert( G) def • S z  = { z  γ | γ ∈ G s } ⊆ V δ≤1 ( v );  • any edge ∈ Edge( G) that abuts to two distinct elements of S z  [hence is fixed by the action, determined by s| I G , of some open subgroup of I G — cf. [NodNon], Proposition 3.9, (ii)] necessarily abuts to v   then G s ⊆ D y  . is nonempty. If the action of G s fixes some y  ∈ VCN( G), Thus, we may assume without loss of generality that the action of G s  In particular, it follows that the does not fix any element of VCN( G). [nonempty!] sets S z  and S δ — both of which are clearly preserved by the action of G s — are of cardinality ≥ 2. In a similar vein, S δ \ (S δ ∩ S z  ) is either empty or of cardinality ≥ 2. Moreover, the latter case contradicts [NodNon], Lemma 1.8. Thus, we conclude that S δ ⊆ S z  , which, by the definition of S z  and S δ , implies that S δ = S z  , i.e., that, for any two distinct z  1 , z  2 ∈ S z  , there exists a [unique, by [NodNon], Lemma 1.8]  such that V( e  ∈ Edge( G) e ) = { z 1 , z  2 }. But, in light of the definition   contains an of S δ , this implies that S z  = 2, and hence that Edge( G) element fixed by the action of G s — a contradiction! This completes  hence also the proof of assertion (iii) in the case where z  ∈ Vert( G), of assertion (iii). Assertion (iv) follows immediately from a similar argument to the argument applied in the proof of assertion (iii). This completes the proof of Lemma 1.11.  30 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 1.12 (Triviality via passage to abelianizations). Let G and J be profinite groups and φ : J → G a continuous homomorphism. Then the following hold: (i) Let γ ∈ G be such that, for every open subgroup H ⊆ G of G that contains γ, the image of γ in H ab is trivial. Then γ is trivial. (ii) Suppose that, for every open subgroup H ⊆ G of G, the com- φ posite φ −1 (H) → H  H ab is trivial. Then φ is trivial. Proof. First, we verify assertion (i). Assume that γ is nontrivial. Then it is immediate that there exists a normal open subgroup N ⊆ G of G such that γ ∈ N . Write H ⊆ G for the closed subgroup of G topologically generated by N and γ. Then the image of γ in the abelian quotient H  H/N is nontrivial. This completes the proof of assertion (i). Assertion (ii) follows immediately from assertion (i). This completes the proof of Lemma 1.12.  Theorem 1.13 (The combinatorial section conjecture for outer representations of ENN-type). Let Σ be a nonempty set of prime numbers, G a semi-graph of anabelioids of pro-Σ PSC-type, G a profi- nite group, and ρ : G → Aut(G) a continuous homomorphism that is of ENN-type for a conducting subgroup I G ⊆ G [cf. Definition 1.7, (i)]. Write Π G for the [pro-Σ] fundamental group of G and Π tp G for the tempered fundamental group of G [cf. [SemiAn], Example 2.10; the dis- cussion preceding [SemiAn], Proposition 3.6]. [Thus, we have a natural outer injection Π tp G → Π G — cf. [CbTpIII], Lemma 3.2, (i); the proof of def out [CbTpIII], Proposition 3.3, (i), (ii).] Write Π G = Π G  G [cf. the dis- def out tp cussion entitled “Topological groups” in [CbTpI], §0]; Π tp G = Π G  G; G  → G, G  tp → G for the universal pro-Σ and pro-tempered coverings of G corresponding to Π G , Π tp G ; VCN(−) for the set of vertices, cusps, and nodes of the underlying [pro-]semi-graph of a [pro-]semi-graph of an- abelioids [cf. Definition 1.1, (i)]. Thus, we have a natural commutative diagram 1 −−−→ Π tp −−−→ Π tp −−−→ G −−−→ 1 ⏐ G  ⏐ G ⏐  ⏐    1 −−−→ Π G −−−→ Π G −−−→ G −−−→ 1 — where the horizontal sequences are exact; the vertical arrows are  tp  outer injections; Π tp G acts naturally on G ; Π G acts naturally on G. Then the following hold: (i) Suppose that ρ is l-cyclotomically full [cf. Definition 1.7, (ii)] for some l ∈ Σ. Let s : G → Π G be a continuous section of the natural surjection Π G  G. Then, relative to the action of COMBINATORIAL ANABELIAN TOPICS IV 31  via conjugation of VCN-subgroups, the image Π G on VCN( G)  of s stabilizes some element of VCN( G). tp (ii) Let s : G → Π G be a continuous section of the natural surjec- tp  tp tion Π tp G  G. Then, relative to the action of Π G on VCN( G ) via conjugation of VCN-subgroups [cf. Definition 1.9], the im- age of s stabilizes some element of VCN( G  tp ). (iii) Write Sect(Π G /G) for the set of Π G -conjugacy classes of con- tinuous sections of the natural surjective homomorphism Π G  tp G and Sect(Π tp G /G) for the set of Π G -conjugacy classes of continuous sections of the natural surjective homomorphism Π tp G  G. Then the natural map Sect(Π tp G /G) −→ Sect(Π G /G) is injective. If, moreover, ρ is l-cyclotomically full for some l ∈ Σ, then this map is bijective. Proof. First, we verify assertion (i). Let us first observe that by replac- ing I G by a suitable open subgroup of I G and G by the pro-l completion of the finite étale covering of G determined by a varying normal open subgroup H ⊆ Π G such that s(G) ⊆ H [cf. Lemma 1.8; [CbTpIII], Lemma 1.5], it follows immediately from the well-known fact that a projective limit of nonempty finite sets is nonempty that we may as- sume without loss of generality that Σ = {l}. Next, let us observe that we may assume without loss of generality that G has at least one node. In particular, it follows immediately from Lemma 1.11, (iii), that, to verify assertion (i), by replacing Π G by a suitable open subgroup of Π G , we may assume without loss of generality — i.e., by arguing as in the discussion entitled “Curves” in [AbsTpII], §0 — that the pro-l completion Π G of the topological fundamental group of the underlying semi-graph G of G is a free pro-l group of rank ≥ 2, hence, in particular, center-free. Then we claim that the following assertion holds: Claim 1.13.A: For every connected finite étale Galois subcovering H → G of G  → G that determines a nor- mal open subgroup of Π G , the action of I G on H, via s, fixes an element of VCN(H). To verify Claim 1.13.A, let us observe that, by replacing H by G [cf. [CbTpIII], Lemma 1.5], we may assume without loss of generality that H = G. Next, let us observe that since the underlying semi-graph G of G is finite, the continuous action of G on G factors through a finite def quotient G  Q, i.e., by a normal open subgroup of G. Write Π G//Q = out Π G  Q [i.e., notation which is well-defined since Π G is center-free — cf. the discussion entitled “Topological groups” in [CbTpI], §0; the notational conventions of Lemma 1.6, in the case where “Σ” is taken 32 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI to be {l}]. Thus, we obtain a commutative diagram 1 −−−→ Π G −−−→ Π G −−−→ G −−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐    1 −−−→ Π G −−−→ Π G//Q −−−→ Q −−−→ 1 — where the horizontal sequences are exact, and the vertical arrows are surjective. Write I G  I Q for the [finite] quotient of I G determined by def def the quotient G  Q, N G = Ker(G  Q), and N I = Ker(I G  I Q ). Now let us observe that (a) since Q is finite, it is immediate that N G , N I are open in G, I G , respectively, and, moreover, (b) it follows from the definition of the term “ENN-type” that, by replacing G  Q by a suitable quotient of Q if necessary, we may assume without loss of generality that the quotient {l}-ab-free [cf. the I G  I Q factors through the quotient I G  I G discussion entitled “Groups” in §0], hence is cyclic of order a power of l. s Next, let us observe that the composite N G → G → Π G  Π G//Q determines a commutative diagram N I −→ N G | | ↓ ↓ Π G == Π G — where the upper horizontal arrow is the natural inclusion. Now we claim that the following assertion holds: Claim 1.13.B: The left-hand vertical arrow N I → Π G of the above diagram is the trivial homomorphism. Indeed, let H ⊆ Π G be an open subgroup and write N I,H ⊆ N I and N G,H ⊆ N G for the open subgroups obtained by forming the inverse image of H ⊆ Π G via the vertical arrows of the above commutative diagram. Thus, N G,H normalizes N I,H ; the action of N G,H on H by conjugation induces the trivial action of N G,H on H ab . Next, let us observe that since H ab is a free Z l -module, the left-hand vertical arrow {l}-ab-free → H ab of under consideration determines a homomorphism N I,H free Z l -modules of finite rank [cf. Definition 1.7, (i)], which is N G,H - equivariant [with respect to the actions of N G,H by conjugation]. On the other hand, since the action of N G,H on H ab is trivial, the N G,H - {l}-ab-free equivariant homomorphism N I,H → H ab factors through a quo- {l}-ab-free tient of N I,H on which N G,H acts trivially. Thus, since ρ is l- {l}-ab-free cyclotomically full, and N G,H acts on N I,H via the cyclotomic character [cf. Definition 1.7, (i); [CbTpI], Lemma 5.2, (ii)], we con- {l}-ab-free clude that the N G,H -equivariant homomorphism N I,H → H ab is COMBINATORIAL ANABELIAN TOPICS IV 33 trivial. In particular, Claim 1.13.B follows from Lemma 1.12, (ii). This completes the proof of Claim 1.13.B. Next, let us observe that it follows immediately from Claim 1.13.B that the section s determines a section of the natural surjection def pr 2 Π G//I Q = Π G//Q × Q I Q  I Q . Thus, it follows immediately from the resp’d portion of Lemma 1.6, (iv), together with the observation (b) discussed above [cf. also Re- mark 1.13.1 below], that Claim 1.13.A holds. This completes the proof of Claim 1.13.A. Now by allowing the subcovering H in Claim 1.13.A to vary, we conclude immediately [from the well-known fact that a projective limit of nonempty finite sets is nonempty] that s(I G ) stabilizes some element  Thus, it follows from Lemma 1.11, (iii), that the image of VCN( G).  This completes the proof of s(G) stabilizes some element of VCN( G). assertion (i). Assertion (ii) follows, by applying [NodNon], Proposition 3.9, (i), from a similar argument to the argument applied to prove [SemiAn], Theorems 3.7, 5.4. That is to say, instead of considering “subjoints” [i.e., paths of length 2] as in the proof of [SemiAn], Theorem 3.7, the content of [NodNon], Proposition 3.9, (i), requires us to consider paths of length 3. This completes the proof of assertion (ii). Finally, we verify assertion (iii). Let s, t : G → Π tp G be sections of tp the surjection Π G  G such that there exists an element γ ∈ Π G such s that the composite s  : G → Π tp G → Π G is the conjugate by γ ∈ Π G of t tp the composite  t : G → Π G → Π G . Thus, it follows from assertion (ii) [applied to both s and t] that there exist elements y  , z  ∈ VCN( G  tp )  for the image of z  by the action such that if we write z  γ ∈ VCN( G) of γ, then s  stabilizes both y  and z  γ . In particular, we conclude from Lemma 1.11, (ii), that the distance between y  and z  γ is finite, hence that, for each subcovering H → G of G  tp → G that arises from an  and z  γ in open subgroup of Π tp G , the distance between the images of z tp H is finite, which implies that γ ∈ Π G . This completes the proof of the injectivity portion of assertion (iii). Since [one verifies immediately  lies in the Π G -orbit of an element of that] every element of VCN( G) VCN( G  tp ), the final portion of assertion (iii) follows immediately from assertion (i). This completes the proof of Theorem 1.13.  Remark 1.13.1. We observe in passing, with regard to the application of Lemma 1.6, (iv), in the proof of Theorem 1.13, (i), that, in fact, Lemma 1.6, (iv), is only applied in the case where the group “G” of Lemma 1.6 is cyclic and of order a power of l. That is to say, we only 34 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI apply Lemma 1.6, (iv), in the case that, as discussed in Remark 1.6.2, (i), admits a relatively simple proof. Corollary 1.14 (A combinatorial version of the Grothendieck conjecture for outer representations of ENN-type). Let Σ be a nonempty set of prime numbers; G, H semi-graphs of anabelioids of ∼ pro-Σ PSC-type; G G , G H profinite groups; β : G G → G H a continu- ous isomorphism; ρ G : G G → Aut(G), ρ H : G H → Aut(H) continuous homomorphisms that are of ENN-type for conducting subgroups I G G ⊆ G G , I G H ⊆ G H [cf. Definition 1.7, (i)] such that β(I G G ) = I G H ; l ∈ Σ a prime number such that ρ G and ρ H are l-cyclotomically full [cf. Definition 1.7, (ii)]. Suppose further that ρ G is verticially quasi- split [cf. Definition 1.7, (i)]. Write Π G , Π H for the [pro-Σ] funda- ∼ mental groups of G, H, respectively. Let α : Π G → Π H be a continuous isomorphism such that the diagram G G ρ G −→ Aut(G) −→ Out(Π G ) | | ↓ β ↓ G H ρ H −→ Aut(H) −→ Out(Π H ) — where the right-hand vertical arrow is the isomorphism obtained by conjugating by α — commutes. Then α is graphic [cf. [CmbGC], Definition 1.4, (i)]. Proof. First, let us observe that by [CmbGC], Corollary 2.7, (i), it follows from our assumption that ρ G and ρ H are l-cyclotomically full ∼ that α : Π G → Π H is group-theoretically cuspidal. Thus, by applying [CmbGC], Proposition 1.5, (ii); [NodNon], Lemma 1.14, we conclude that it suffices to verify that α is group-theoretically verticial under the additional assumption that G and H are noncuspidal. Write Π G G , Π G H for the profinite groups “Π G ” [cf. Theorem 1.13] associated to ρ G , ρ H . Then it follows immediately from our assumption that ρ G is verticially quasi-split that we may assume, after possibly replacing G G and G H by corresponding open subgroups, that there exists a section s G : G G → Π G G such that the image of the restriction of s G to I G G com- mutes with some verticial subgroup of Π G . In particular, s G satisfies the conditions imposed on the section “s : G → Π G ” in Lemma 1.11, (ii), for some nonempty subset “S”. Moreover, it follows from The- ∼ orem 1.13, (i), that the isomorphism Π G G → Π G H determined by α and β maps s G to a section s H : G H → Π G H that is contained in the normalizer in Π G H of a VCN-subgroup of Π H . In particular, after possibly replacing G G and G H by corresponding open subgroups, we may assume [cf. [CmbGC], Proposition 1.2, (ii); [NodNon], Remark COMBINATORIAL ANABELIAN TOPICS IV 35 2.7.1] that the image of the restriction of s H to I G H commutes with some nontrivial verticial element of Π H [cf. [CbTpII], Definition 1.1]. Thus, by restricting these sections s G , s H to the respective conducting subgroups and forming appropriate centralizers [cf. [NodNon], Lemma 3.6, (i), applied to the restriction of s G to I G G ], we conclude from the assumption that β is compatible with the respective conducting sub- ∼ groups that α : Π G → Π H maps some nontrivial verticial element of Π G to a nontrivial verticial element of Π H . In particular, it follows from the implication (3) ⇒ (1) of [CbTpII], Theorem 1.9, (i), that α is group-theoretically verticial, as desired.  Remark 1.14.1. It is not difficult to verify that the assumption in the statement of Corollary 1.14 that β(I G G ) = I G H cannot be omitted. Indeed, if one omits this assumption, then a counterexample to the graphicity asserted in Corollary 1.14 may be obtained as follows: Let J be a semi-graph of anabelioids of pro-Σ PSC-type and e G , e H distinct nodes of J . Write G (respectively, H) for the semi-graph of anabe- lioids of pro-Σ PSC-type J Node(J )\{e G } (respectively, J Node(J )\{e H } ) obtained by deforming the nodes of J that are  = e G (respectively,  = e H ) [cf. [CbTpI], Definition 2.8]; I G G (respectively, I G H ) for the [nec- essarily normal — cf. [CbTpI], Theorem 4.8, (i), (v)] closed subgroup of Aut |{e G ,e H }| (J ) [cf. [CbTpI], Definition 2.6, (i)] generated by the profi- nite Dehn twists that arise from the direct summand of the direct sum decomposition in the display of [CbTpI], Theorem 4.8, (iv), labeled by e G (respectively, e H ). Next, let G G = G H be a closed subgroup of Aut |{e G ,e H }| (J ) such that • G G = G H contains both I G G and I G H , • the natural inclusion G G = G H → Aut(J ) is l-cyclotomically full for some l ∈ Σ, and, moreover, • if we write ρ G (respectively, ρ H ) for the continuous injection G G → Aut(G) (respectively, G H → Aut(H)) obtained by forming the composite of the natural inclusion G G = G H → Aut |{e G ,e H }| (J ) and the injection Aut |{e G ,e H }| (J ) → Aut(G) (respectively, Aut |{e G ,e H }| (J ) → Aut(H)) [cf. [CbTpI], Propo- sition 2.9, (ii)], then ρ G is verticially quasi-split. [Note that one verifies easily the existence of such a closed subgroup of Aut |{e G ,e H }| (J ) by considering, for instance, a homomorphism G G = G H → Aut(J ) of EPIPSC-type that arises from a suitable stable log curve — cf. also Remark 1.7.1; [CbTpI], Lemma 5.4, (ii); [CbTpI], Proposition 5.6, (ii).] Then if one takes the “α” of Corollary 1.14 to be the outer isomorphism determined by the specialization outer isomorphisms Φ J Node (J )\{e G } , Φ J Node (J )\{e H } [cf. [CbTpI], Definition 2.10] and the “β” of Corollary 1.14 to be the identity isomorphism, then 36 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI one verifies immediately from [CbTpI], Corollary 3.9, (i), and [CbTpI], Corollary 5.9, (iii), that one obtains a counterexample as desired. Let R be a complete discrete valuation ring whose residue character- istic we denote by p [so p may be zero]; K a separable closure of the field of fractions K of R; X log a stable log curve [cf. the discussion entitled “Curves” in [CbTpI], §0] over the log regular log scheme Spec(R) log obtained by equipping Spec(R) with the log structure determined by the maximal ideal m R ⊆ R of R. Suppose, for simplicity, that X log is split, i.e., that the nat- ural action of Gal(K/K) on the dual semi-graph Γ X log associated to def the geometric special fiber of X log is trivial. Write X log = X log × R K; Vert(X log ) (respectively, Cusp(X log ); Node(X log )) for the set of ver- tices (respectively, open edges; closed edges) of Γ X log , i.e., the set of connected components of the complement of the cusps and nodes (respectively, the set of cusps; the set of nodes) of the special fiber of X log ; def VCN(X log ) = Vert(X log )  Cusp(X log )  Node(X log ). Before proceeding, we recall that to each element z ∈ VCN(X log ), one may associate, in a way that is functorial with respect to arbitrary auto- morphisms of the log scheme X log , a discrete valuation that dominates R on the residue field of some point of X, which is closed if and only if z is a cusp. Indeed, this is immediate if z is a vertex, since a vertex corresponds to a prime of height 1 of X . This is also immediate if z is a cusp, since the residue field of the closed point of X that corresponds to z is finite over [the complete discrete valuation field] K, which implies that the discrete valuation of K extends uniquely to a discrete valuation on the residue field of a cusp. Now suppose that z is a node that is locally defined by an equation of the form s 1 s 2 − a, for some a ∈ m R [cf., e.g., the discussion of [CbTpI], Definition 5.3, (ii)]. By descent, we may assume without loss of generality that a admits a square root b in R. Then one associates to z the discrete valuation determined by the exceptional divisor of the blow-up of X at the locus (s 1 , s 2 , b). [One verifies immediately that this construction is compatible with arbitrary automorphisms of X log .] Corollary 1.15 (Fixed points associated to Galois sections). Let Σ be a set of prime numbers; Σ † ⊆ Σ a subset; l ∈ Σ † ; R a complete COMBINATORIAL ANABELIAN TOPICS IV 37 discrete valuation ring of residue characteristic p ∈ Σ † [so p may be zero]; K a separable closure of the field of fractions K of R; X log a stable log curve [cf. the discussion entitled “Curves” in [CbTpI], §0] over the log regular log scheme Spec(R) log obtained by equipping Spec(R) with the log structure determined by the maximal ideal of R. def Write G K = Gal(K/K) for the absolute Galois group of K; I K ⊆ G K def log def for the inertia subgroup of G K ; X log = X log × R K; X K = X log × R K; Δ X log log for the pro-Σ log fundamental group of X K [i.e., the maximal pro-Σ log quotient of the log fundamental group of X K ]; Π X log for the geometrically pro-Σ log fundamental group of X log [i.e., the quotient of the log fundamental group of X log by the kernel of the natural log surjection from the log fundamental group of X K onto Δ X log ]. Thus, we have a natural exact sequence of profinite groups 1 −→ Δ X log −→ Π X log −→ G K −→ 1.  log → X log for the profinite log étale covering of X log corre- Write X sponding to Π X log . If Y log → X log is a finite connected subcovering of  log → X log that admits a stable model Y log over the normalization R Y X of R in Y , then let us write Γ Y log for the dual semi-graph determined by the geometric special fiber of Y log over R Y ; Vert(Y log ) (respectively, Cusp(Y log ); Node(Y log )) for the set of vertices (respectively, open edges; closed edges) of Γ Y log , i.e., the set of connected components of the com- plement of the cusps and nodes (respectively, the set of cusps; the set of nodes) of the geometric special fiber of Y log over R Y ; def Edge(Y log ) = Cusp(Y log )  Node(Y log ); def VCN(Y log ) = Vert(Y log )  Edge(Y log );  log ) = lim VCN(Y log ) VCN( X ←− — where the projective limit is over all finite connected subcoverings  log → X log as above, and, moreover, for each finite Y log → X log of X  log → X log that admits a stable connected subcovering Y 1 log → X log of X log model Y 1 over the normalization of R in Y 1 , the transition map for  log → Y 1 log that admits a finite connected subcovering Y 2 log → Y 1 log of X a stable model Y 2 log over the normalization of R in Y 2 is defined, for z ∈ VCN(Y 2 log ), as follows: def 38 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI • If the connected component/cusp/node corresponding to z maps, via the extension Y 2 log → Y 1 log of Y 2 log → Y 1 log [cf., e.g., [ExtFam], Theorem C], to a cusp or node of the geometric special fiber of Y 1 , then the image of z ∈ VCN(Y 2 log ) in VCN(Y 1 log ) is defined to be the element of Edge(Y 1 log ) corresponding to the cusp or node. • If the generic point of the connected component/cusp/node cor- responding to z maps, via the extension Y 2 log → Y 1 log of Y 2 log → Y 1 log , to a point of the geometric special fiber of Y 1 that is neither a cusp nor node, then the image of z ∈ VCN(Y 2 log ) in VCN(Y 1 log ) is defined to be the element of Vert(Y 1 log ) corre- sponding to the connected component on which the point lies.  log ), and Y log → X log is a finite connected subcovering If z  ∈ VCN( X  log → X log that admits a stable model Y log over the normalization of X of R in Y , then let us write z  (Y log ) ∈ VCN(Y log ) for the element of VCN(Y log ) determined by z  . Let H ⊆ G K be a closed subgroup such that the image of def I H = H ∩ I K ⊆ I K Σ † Σ † via the natural surjection I K  I K to the pro-Σ † completion I K of I K † Σ is an open subgroup of I K and s : H −→ Π X log a section of the restriction to H ⊆ G K of the above exact sequence 1 → Δ X log → Π X log → G K → 1. Then the following hold: (i) If we write Δ † X log for the maximal pro-Σ † quotient of Δ X log and regard, via the specialization outer isomorphism with respect to X log , the pro-Σ † group Δ † X log as the [pro-Σ † ] fundamental group of the semi-graph of anabelioids of pro-Σ † PSC-type de- termined by the geometric special fiber of the stable model X log [cf. [CmbGC], Example 2.5], then the natural outer Galois ac- tion H −→ Out(Δ † X log ) determined by the above exact sequence is of EPIPSC-type for the conducting subgroup I H ⊆ H [cf. Definition 1.7, (i)]. If, moreover, H is l-cyclotomically full, i.e., the image of H ⊆ G K via the l-adic cyclotomic character on G K is open, then the above outer Galois action is l-cyclotomically full [cf. Definition 1.7, (ii)]. (ii) Suppose that the residue field of R is countable. Let z  ∈  log ) and S = {Y log → X log } a cofinal system consisting VCN( X  log → X log such of finite Galois subcoverings Y log → X log of X that Y log admits a split stable model over the normalization R Y of R in Y . Then there exist a valuation v z  on the residue COMBINATORIAL ANABELIAN TOPICS IV 39  of X  log [i.e., field of some point of the underlying scheme X a bounded multiplicative seminorm — cf., e.g., [Brk1], §1.1, §1.2] and a countably indexed cofinal subsystem S  of S such that if Z log → X log is a member of S  , then, as Y log → X log ranges over the members of S  that lie over Z log , the discrete valuations on residue fields of points of the underlying scheme Z of Z log determined by the elements z  (Y log ) ∈ VCN(Y log ) [cf. the discussion preceding the present Corollary 1.15] converge in the “Berkovich space topology” — i.e., as bounded mul- tiplicative seminorms — to the valuation on the residue field of some point of Z determined by v z  .  log ) for the subset consisting of el- (iii) Write Stab(s) ⊆ VCN( X  log ) such that the image of s stabilizes ements z  ∈ VCN( X z  . Suppose that H is l-cyclotomically full [cf. (i)]. Then it holds that Stab(s)  = ∅. In particular, if z  ∈ Stab(s), and the residue field of R is countable, then the image of s lies in the decomposition group of any valuation v z  as in (ii).  log → (iv) Let Y log → X log be a finite connected subcovering of X log X that admits a stable model over the normalization R Y of R in Y ; z  1 , z  2 ∈ Stab(s) [cf. (iii)]. Then one of the following four [mutually exclusive] conditions is satisfied: z 1 (Y log ), z  2 (Y log )) ≤ (a) z  1 (Y log ), z  2 (Y log ) ∈ Vert(Y log ), and δ( 2 [cf. Definition 1.1, (iii)]. (b) z  1 (Y log ), z  2 (Y log ) ∈ Edge(Y log ), and, moreover, V( z 1 (Y log )) ∩V( z 2 (Y log ))  = ∅. (c) z  1 (Y log ) ∈ Vert(Y log ), z  2 (Y log ) ∈ Edge(Y log ), and, more- z 1 (Y log ))∩V( z 2 (Y log ))  = ∅ [cf. Definition 1.10, over, V δ≤1 ( (i)]. (d) z  1 (Y log ) ∈ Edge(Y log ), z  2 (Y log ) ∈ Vert(Y log ), and, more- over, V( z 1 (Y log )) ∩ V δ≤1 ( z 2 (Y log ))  = ∅. (v) In the situation of (iv), suppose, moreover, that the following assertion — i.e., concerning “resolution of nonsingulari- ties” [cf. Remark 1.15.1 below] — holds: († RNS ): Let Y log → X log be a finite connected sub-  log → X log that admits a stable model covering of X Y log over R Y and y ∈ Y a node of Y. Then there exists a finite connected subcovering Z log → Y log of  log → Y log that admits a stable model Z log over R Z X such that the fiber over y of the morphism Z → Y determined by Z log → Y log is not finite. 40 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI  log → Then every finite connected subcovering Y log → X log of X log X that admits a stable model over R Y satisfies one of the following four [mutually exclusive] conditions: (a  ) z  1 (Y log ), z  2 (Y log ) ∈ Vert(Y log ), and z  1 (Y log ) = z  2 (Y log ). (b  ) z  1 (Y log ), z  2 (Y log ) ∈ Edge(Y log ), and, moreover, V( z 1 (Y log )) log ∩V( z 2 (Y ))  = ∅.  (c ) z  1 (Y log ) ∈ Vert(Y log ), z  2 (Y log ) ∈ Edge(Y log ), and, more- z 2 (Y log )). over, z  1 (Y log ) ∈ V( (d  ) z  1 (Y log ) ∈ Edge(Y log ), z  2 (Y log ) ∈ Vert(Y log ), and, more- z 1 (Y log )). over, z  2 (Y log ) ∈ V( tp (vi) Write Δ X log for the Σ-tempered fundamental group of log [cf. [CbTpIII], Definition 3.1, (ii)]; Π tp for the geo- X K X log metrically Σ-tempered fundamental group of X log [i.e., the quotient of the tempered fundamental group of X log by the kernel of the natural surjection from the tempered fundamen- log tal group of X K onto Δ tp ]. Thus, we have a natural exact X log sequence of topological groups 1 −→ Δ tp −→ Π tp −→ G K −→ 1. X log X log Write Sect(Π X log /H) for the set of Δ X log -conjugacy classes of continuous sections of the restriction to H ⊆ G K of the natural surjection Π X log  G K and Sect(Π tp /H) for the set of Δ tp - X log X log conjugacy classes of continuous sections of the restriction to  G K . Then the H ⊆ G K of the natural surjection Π tp X log natural map Sect(Π tp /H) −→ Sect(Π X log /H) X log is injective. If, moreover, H is l-cyclotomically full [cf. (i)], then this map is bijective. Proof. Assertion (i) follows immediately from the definition of the term “IPSC-type” [cf. [NodNon], Definition 2.4, (i)], together with the well- known structure of the maximal pro-Σ † quotient of I K . Next, we verify assertion (ii). Let us first observe that it follows immediately from our countability assumption on the residue field of R that the following three assertions hold: • If Y log → X log is a member of S, and z  (Y log ) ∈ Cusp(Y log ), then the function field of Y admits a subset which is countable and dense, i.e., with respect to the topology determined by the discrete valuation determined by the element z  (Y log ) ∈ VCN(Y log ). • If Y log → X log is a member of S, and z  (Y log ) ∈ Cusp(Y log ), then the normalization R Y of R in Y admits a subset which COMBINATORIAL ANABELIAN TOPICS IV 41 is countable and dense, i.e., with respect to the topology de- termined by the discrete valuation determined by the element z  (Y log ) ∈ VCN(Y log ). • There exists a countably indexed cofinal subsystem of S [cf., e.g., [AbsTpII], Lemma 2.1]. Thus, assertion (ii) follows immediately, by applying a standard ar- gument involving Cantor diagonalization, from the well-known [local] compactness of Berkovich spaces [cf., e.g., [Brk1], Theorem 1.2.1]. Here, we recall in passing that this compactness is, in essence, a conse- quence of the compactness of a product of copies of the closed interval [0, 1] ⊆ R. This completes the proof of assertion (ii). We refer to Theorem A.7 in Appendix for another approach to proving assertion (ii). Assertion (iii) follows immediately from the observation that, by ap- plying Theorem 1.13, (i) [cf. also Remark 1.7.1; assertion (i) of the present Corollary 1.15; [CmbGC], Proposition 1.2, (i)], together with the well-known fact that a projective limit of nonempty finite sets is  log → X log , nonempty, to the various finite connected subcoverings of X  log fixes some one may conclude that the action of G K , via s, on X  log ) of VCN( X  log ). [Here, we note that when one element z  s ∈ VCN( X applies Theorem 1.13, (i), to the various finite connected subcoverings  log → X log , the conducting subgroup “I G ” of Theorem 1.13, (i), of X must be allowed to vary among suitable open subgroups of the origi- nal conducting subgroup I G .] Assertion (iv) follows immediately [cf. also Remark 1.7.1; assertion (i) of the present Corollary 1.15] from Lemma 1.11, (ii). Next, we verify assertion (v). Let us first observe that it follows immediately from assertion (iv) that if Y log → X log is a finite connected  log → X log that admits a stable model over R Y , then subcovering of X log z  1 (Y ) and z  2 (Y log ) lie in a connected sub-semi-graph Γ ∗ of Γ Y log such that VCN(Γ ∗ )  = Vert(Γ ∗ )  + Edge(Γ ∗ )  ≤ 3 + 2 = 5. Now one verifies immediately that this uniform bound “5” implies that there exists a cofinal system S = {Y log → X log } consisting of finite  log → X log such that Y log admits Galois subcoverings Y log → X log of X a stable model over R Y , and, moreover, Γ Y log admits a connected sub- semi-graph Γ ∗ Y log such that • z  1 (Y log ) and z  2 (Y log ) lie in Γ ∗ Y log ; • VCN(Γ ∗ Y log )  ≤ 5; • the semi-graphs Γ ∗ Y log map isomorphically to one another as one varies Y log → X log . 42 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI def Write V ∗ (Y log ) = Vert(Γ ∗ Y log ). Then it follows immediately from asser- tion (iv) that, to complete the verification of assertion (v), it suffices to verify that the following assertion holds: Claim 1.15.A: V ∗ (Y log )  ≤ 1. Indeed, suppose that V ∗ (Y log )  ≥ 2. Then it follows immediately that there exists a compatible system of nodes e(Y log ) of Γ ∗ Y log [i.e., compat- ible as one varies Y log → X log in S], each of which abuts to distinct vertices v α (Y log ), v β (Y log ) of Γ ∗ Y log . [Thus, one may assume that the vertices v α (−) (respectively, v β (−)) form a compatible system of ver- tices.] But this implies that for every Z log → X log in S that lies over Y log → X log in S, if we write Y log , Z log for the respective stable mod- els of Y log , Z log [so the morphism Z log → Y log extends to a morphism Z log → Y log — cf., e.g., [ExtFam], Theorem C], then the inverse im- age in Z log of the node e(Y log ) admits at least one isolated point [i.e., e(Z log )], hence [since the covering Z log → Y log is Galois] the entire in- verse image in Z log of e(Y log ) is of dimension zero. On the other hand, this contradicts the assertion († RNS ) in the statement of assertion (v). This completes the proof of assertion (v). Finally, we verify assertion (vi). The injectivity portion of assertion (v) follows immediately from the injectivity portion of Theorem 1.13, (iii) [cf. also Remark 1.7.1; assertion (i) of the present Corollary 1.15],  log → X log , applied to the various finite connected subcoverings of X † where we take the “Σ” of Theorem 1.13 to be Σ [cf. also the fact that, in the notation of Theorem 1.13, “Π tp G ” is dense in “Π G ” in the profinite topology]. Here, we note that • when one applies Theorem 1.13, (iii), to the various finite con-  log → X log , the conducting subgroup nected subcoverings of X “I G ” of Theorem 1.13, (iii), must be allowed to vary among suitable open subgroups of the original conducting subgroup I G , and that • it follows immediately from the final portion of Lemma 1.11, (iv), that the resulting conjugacy indeterminacies that occur at various subcoverings are uniquely determined up to profinite centralizers of the sections that appear, hence converge in Δ tp X log [i.e., if one passes to an appropriate subsequence of the system of subcoverings under consideration]. If H is l-cyclotomically full, then the surjectivity of the map /H) → Sect(Π X log /H) Sect(Π tp X log follows formally [cf. the proof of the final portion of Theorem 1.13, (iii)] from the nonemptiness verified in assertion (iii). This completes the proof of assertion (vi).  COMBINATORIAL ANABELIAN TOPICS IV 43 Remark 1.15.1. It follows from [Tama2], Theorem 0.2, (v), that if K is of characteristic zero, the residue field of R is algebraic over F p , and Σ = Primes, then the assertion († RNS ) in the statement of Corol- lary 1.15, (v), holds. Remark 1.15.2. (i) Corollary 1.15, (iii), (v) [cf. also [SemiAn], Lemma 5.5], may be regarded as a generalization of the Main Result of [PS]. These results are obtained in the present paper [cf. the proof of Theorem 1.13, (i)] by, in essence, combining, via a simi- lar argument to the argument applied in the tempered case treated in [SemiAn], Theorems 3.7, 5.4 [cf. also the proof of Theorem 1.13, (ii), of the present paper], the uniqueness re- sult given in [NodNon], Propositions 3.8, (i); 3.9, (i), (ii), (iii) [cf. the proof of Lemma 1.11, (ii)], with the existence of fixed points of actions of finite groups on graphs that follows as a consequence of the classical fact that [discrete or pro-Σ] free groups are torsion-free [cf. Remarks 1.6.2, 1.13.1; the proof of Lemma 1.6, (ii)]. One slight difference between the profinite and tempered cases is that, whereas, in the tempered case, it follows from the discreteness of the fundamental groups of graphs that appear that the actions of profinite groups on uni- versal coverings of such graphs necessarily factor through fi- nite quotients, the corresponding fact in the profinite case is obtained as a consequence of the fact that, under a suitable assumption on the cyclotomic characters that appear, any ho- momorphism from a “positive slope” module to a torsion-free “slope zero” module necessarily vanishes [cf. the proof of Claim 1.13.B in Theorem 1.13, (i)]. That is to say, in a word, these results are obtained in the present paper as a consequence of abstract considerations concerning abstract profi- nite groups acting on abstract semi-graphs that may, for instance, arise as dual semi-graphs of geo- metric special fibers of stable models of curves that appear in scheme theory, but, a priori, have nothing to do with scheme theory. This a priori irrelevance of scheme theory to such abstract considerations is reflected both in the variety of the results obtained in the present §1 as consequences of Theorem 1.13, as well as in the generality of Corollary 1.15. This approach contrasts quite substantially with the approach of [PS], i.e., where the main results are derived as a consequence of highly scheme-theoretic considerations concerning stable curves over complete discrete valuation rings, in which the theory of the 44 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Brauer group of the function field of such a curve plays a central role [cf. [PS], §4]. (ii) The essential equivalence between the issue of considering val- uations fixed by Galois actions and the issue of considering vertices or edges of associated dual semi-graphs fixed by Ga- lois actions may be seen in the well-known functorial homotopy equivalence between the Berkovich space associated to a stable curve over a complete discrete valuation ring and the associated dual graph [cf. [Brk2], Theorems 8.1, 8.2]. Moreover, the issue of convergence of [sub]sequences of valuations fixed by Galois actions is an easy consequence of the well-known [local] com- pactness of Berkovich spaces [cf. the proof of Corollary 1.15, (ii); [Brk1], Theorem 1.2.1], i.e., in essence, a consequence of the well-known compactness of a product of copies of the closed interval [0, 1] ⊆ R. That is to say, there is no need to consider the quite complicated [and, at the time of writing, not well understood!] structure of inductive limits of local rings, as dis- cussed in [PS], §1.6. Remark 1.15.3. Recall that in Corollary 1.15, (ii), and the final por- tion of Corollary 1.15, (iii), we assume that the residue field of R is countable. In fact, however, it is not difficult to see that, in the situ- ation of Corollary 1.15, there exists a complete discrete valuation ring R † that is dominated by R, and whose residue field is countable such that • the smooth log curve X log , • the closed subgroup H ⊆ G K , and • the section s : H → Π X log descend to the field of fractions of R † . Indeed, let us first observe that since the moduli stack of pointed stable curves of a given type over Z is of finite type over Z, there exists a complete discrete valuation ring R ‡ that is dominated by R, and whose residue field is countable such that the smooth log curve X log descends to the field of fractions of R ‡ . Next, let us observe that since [cf., e.g., [CanLift], Proposition 2.3, (ii)] the geometric fundamental group “Δ X log ” associated to the smooth log curve X log [i.e., over the field of fractions of R] is naturally isomorphic to the geometric fundamental group “Δ X log ” associated to the descended smooth log curve [i.e., over the field of fractions of R ‡ ], it follows that both of these geometric fundamental groups are topolog- ically finitely generated [cf., e.g., [MT], Proposition 2.2, (ii)], and hence that there exists a countably indexed open basis . . . ⊆ U n+1 ⊆ U n ⊆ . . . ⊆ U 2 ⊆ U 1 ⊆ U 0 = Δ X log COMBINATORIAL ANABELIAN TOPICS IV 45 of characteristic open subgroups of Δ X log . In particular, there exists a complete discrete valuation ring R † that is dominated by R, and whose residue field is countable such that, for each positive integer n, the finite collection of finite étale coverings [which are defined by means of finitely many polynomials, with finitely many coefficients] corresponding to • the finite quotient Π X log  Q n determined by the image of the composite of the conjugation action Π X log → Aut(Δ X log ) and the natural homomorphism Aut(Δ X log ) → Aut(Δ X log /U n ) and • the subgroup H n ⊆ Q n obtained by forming the image of the composite of the section s : H → Π X log and the natural surjec- tive homomorphism Π X log  Q n descends to the field of fractions of R † . 46 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 2. Discrete combinatorial anabelian geometry In the present §2, we introduce the notion of a semi-graph of temper- oids of HSD-type [i.e., “hyperbolic surface decomposition type” — cf. Definition 2.3, (iii)] and discuss discrete versions of the profinite results obtained in [NodNon], [CbTpI], [CbTpII], [CbTpIII]. A semi-graph of temperoids of HSD-type arises naturally from a decomposition [satis- fying certain properties] of a hyperbolic topological surface and may be regarded as a discrete analogue of the notion of a semi-graph of anabelioids of PSC-type. The main technical result of the present §2 is Theorem 2.15, one immediate consequence of which is the following [cf. Corollary 2.19]: An isomorphism of groups between the discrete funda- mental groups of a pair of semi-graphs of temperoids of HSD-type arises from an isomorphism between the semi-graphs of temperoids of HSD-type if and only if the induced isomorphism between profinite comple- tions of fundamental groups arises from an isomor- phism between the associated semi-graphs of anabe- lioids of pro-Primes PSC-type. In the present §2, let Σ be a nonempty set of prime numbers. Definition 2.1. (i) We shall refer to as a semi-graph of temperoids G a collection of data as follows: • a semi-graph G [cf. the discussion at the beginning of [SemiAn], §1], • for each vertex v of G, a connected temperoid G v [cf. [SemiAn], Definition 3.1, (ii)], • for each edge e of G, a connected temperoid G e , together with, for each branch b ∈ e abutting to a vertex v, a mor- phism of temperoids b ∗ : G e → G v [cf. [SemiAn], Definition 3.1, (iii)]. We shall refer to a semi-graph of temperoids whose underlying semi-graph is connected as a connected semi-graph of temper- oids. Given two semi-graphs of temperoids, there is an evident notion of (1-)morphism [cf. [SemiAn], Definition 2.1; [SemiAn], Remark 2.4.2] between semi-graphs of temperoids. (ii) Let T be a connected temperoid. We shall say that a connected object H of T is Σ-finite if there exists a morphism J → H in T such that J is Galois [hence connected — cf. [SemiAn], Definition 3.1, (iv)], and, moreover, Aut T (J) is a finite group whose order is a Σ-integer [cf. the discussion entitled “Num- bers” in §0]. We shall say that an object H of T is Σ-finite if H is isomorphic to a disjoint union of finitely many connected COMBINATORIAL ANABELIAN TOPICS IV 47 Σ-finite objects. We shall say that an object H of T is a finite object if H is Primes-finite. We shall write T Σ for the connected anabelioid [cf. [GeoAn], Definition 1.1.1] ob- tained by forming the full subcategory of T whose objects are the Σ-finite objects of T . Thus, we have a natural morphism of temperoids [cf. Remark 2.1.1 below] T −→ T Σ . We shall write def T  = T Primes [cf. the discussion entitled “Numbers” in §0]. Finally, we ob- serve that if T = B tp (Π), where Π is a tempered group [cf. [SemiAn], Definition 3.1, (i)], and “B tp (−)” denotes the cate- gory “B temp (−)” of the discussion at the beginning of [SemiAn], §3, then T Σ may be naturally identified with B(Π Σ ), i.e., the connected anabelioid [cf. [GeoAn], Definition 1.1.1; the discus- sion at the beginning of [GeoAn], §1] determined by the pro-Σ completion Π Σ of Π. (iii) Let G be a semi-graph of temperoids [cf. (i)]. Then, by re- placing the connected temperoids “G (−) ” corresponding to the Σ vertices and edges “(−)” by the connected anabelioids “G (−) ” [cf. (ii)], we obtain a semi-graph of anabelioids, which we de- note by G Σ [cf. [SemiAn], Definition 2.1]. Thus, it follows immediately from the various definitions involved that the various mor- Σ ” of (ii) determine a natural morphism phisms “G (−) → G (−) of semi-graphs of temperoids [cf. Remark 2.1.1 below] G −→ G Σ . We shall write G  = G Primes . One verifies easily that if G is a connected semi-graph of temperoids [cf. (i)], then G Σ is a connected semi-graph of anabelioids. (iv) Let G be a connected semi-graph of temperoids [cf. (i)]. Sup- pose that [the underlying semi-graph of] G has at least one vertex. Then we shall write def  B(G) = B( G) def [cf. (iii); the discussion following [SemiAn], Definition 2.1] for the connected anabelioid determined by the connected semi-  graph of anabelioids G. 48 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (v) Let G be a semi-graph of temperoids. Then we shall write Vert(G), Cusp(G), Node(G), Edge(G), VCN(G), V, C, N , E, and δ for the Vert, Cusp, Node, Edge, VCN, V, C, N , E, and δ of Definition 1.1, (i), (ii), (iii), applied to the underlying semi-graph of G. (vi) Let G be a connected semi-graph of temperoids [cf. (i)]. Sup- pose that [the underlying semi-graph of] G has at least one vertex. Then we shall write B tp (G) for the category whose objects are given by collections of data {S v , φ e } — where v (respectively, e) ranges over the elements of Vert(G) (respectively, Edge(G)) [cf. (v)]; for each v ∈ Vert(G), S v is an object of the temperoid G v corresponding to v; for each e ∈ Edge(G), with branches b 1 , b 2 abutting to vertices v 1 , v 2 , ∼ respectively, φ e : ((b 1 ) ∗ ) ∗ S v 1 → ((b 2 ) ∗ ) ∗ S v 2 is an isomorphism in the temperoid G e corresponding to e — and whose morphisms are given by morphisms [in the evident sense] between such collections of data. In particular, the category [i.e., connected anabelioid] B(G) of (iv) may be regarded as a full subcategory B(G) ⊆ B tp (G) of B tp (G). One verifies immediately that any object G  of B tp (G) determines, in a natural way, a semi-graph of temper- oids G  , together with a morphism of semi-graphs of temperoids G  → G. We shall refer to this morphism G  → G as the cov- ering of G associated to G  . We shall say that a morphism of semi-graphs of temperoids is a covering (respectively, finite étale covering) of G if it factors as the post-composite of an isomorphism of semi-graphs of temperoids with the covering of G associated to some object of B tp (G) (respectively, of B(G) (⊆ B tp (G))). We shall say that a covering of G is connected if the underlying semi-graph of the domain of the covering is connected. Remark 2.1.1. Since every profinite group is tempered [cf. [SemiAn], Definition 3.1, (i); [SemiAn], Remark 3.1.1], it follows immediately that a connected anabelioid [cf. [GeoAn], Definition 1.1.1] determines, in a natural way [i.e., by considering formal countable coproducts, as in the discussion entitled “Categories” in [SemiAn], §0], a connected tem- peroid [cf. [SemiAn], Definition 3.1, (ii)]. In particular, a semi-graph of anabelioids [cf. [SemiAn], Definition 2.1] determines, in a natural way, a semi-graph of temperoids [cf. Definition 2.1, (i)]. By abuse of COMBINATORIAL ANABELIAN TOPICS IV 49 notation, we shall often use the same notation for the connected tem- peroid (respectively, semi-graph of temperoids) naturally associated to a connected anabelioid (respectively, semi-graph of anabelioids). Definition 2.2. (i) Let T be a topological space. Then we shall say that a closed subspace of T (respectively, a closed subspace of T ; an open subspace of T ) is a circle (respectively, a closed disc; an open disc) on T if it is homeomorphic to the set { (s, t) ∈ R 2 | s 2 + t 2 = 1 } (respectively, { (s, t) ∈ R 2 | s 2 + t 2 ≤ 1 }; { (s, t) ∈ R 2 | s 2 + t 2 < 1 }) equipped with the topology induced by the topology of R 2 . If D ⊆ T is a closed disc on T , then we shall write ∂D ⊆ D for the circle on T determined by the boundary of D regarded as a two-dimensional topological manifold with boundary [i.e., the closed subspace of D corresponding to the closed subspace { (s, t) ∈ R 2 | s 2 + t 2 = 1 } ⊆ { (s, t) ∈ R 2 | s 2 + def t 2 ≤ 1 }] and D ◦ = D \ ∂D ⊆ D for the open disc on T obtained by forming the complement of ∂D in D. (ii) Let (g, r) be a pair of nonnegative integers. Then we shall say that a pair X = (X, {D i } ri=1 ) consisting of a connected orientable compact topological surface X of genus g and a col- lection of r disjoint closed discs D i ⊆ X of X [cf. (i)] is of HS-type [where the “HS” stands for “hyperbolic surface”] if 2g − 2 + r > 0. (iii) Let X = (X, {D i } ri=1 ) be a pair of HS-type [cf. (ii)]. Then we shall write  r ◦  def U X = X\ D i i=1 [cf. (i)] and refer to U X as the interior of X. We shall refer to a circle on U X determined by some ∂D i ⊆ U X [cf. (i)] as a cusp of U X , or alternatively, X. Write ∂U X ⊆ U X for the union of the cusps of U X ; I X for the group of homeomorphisms ∼ φ : X → X such that φ restricts to the identity on U X . Suppose that Y = (Y , {E i } sj=1 ) is also a pair of HS-type. Then we define ∼ an isomorphism X → Y of pairs of HS-type to be an I X -orbit ∼ of homeomorphisms X → Y such that each homeomorphism ψ ∼ that belongs to the I X -orbit induces a homeomorphism U X → U Y . (iv) Let X = (X, {D i } ri=1 ) be a pair of HS-type [cf. (ii)] and {Y j } j∈J a finite collection of pairs of HS-type. For each j ∈ J, let ι j : U Y j → U X [cf. (iii)] be a local immersion [i.e., a map that restricts to a homeomorphism between some open neigh- borhood of each point of the domain and the image of the 50 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI open neighborhood, equipped with the induced topology, in the codomain] of topological spaces. Then we shall say that a pair ({Y j } j∈J , {ι j } j∈J ) is an HS-decomposition of X if the following conditions are satisfied: (1) U X = j∈J ι j (U Y j ). (2) For any j ∈ J, the complement of the diagonal in U Y j × U X U Y j is a disjoint union of circles, each of which maps home- omorphically, via the two projections to U Y j , to two dis- tinct cusps of U Y j [cf. (iii)]. [Thus, by “Brouwer invariance of domain”, it follows that ι j restricts to an open immer- sion on the complement of the cusps of U Y j .] (3) For any j, j  ∈ J such that j  = j  , every connected com- ponent of U Y j × U X U Y j  projects homeomorphically onto cusps of U Y j and U Y j  . (4) For any [i.e., possibly equal] j, j  ∈ J, we shall refer to a circle of U Y j × U X U Y j  that forms a connected component of U Y j × U X U Y j  as a pre-node [of the HS-decomposition ({Y j } j∈J , {ι j } j∈J )] and to the cusps of U Y j , U Y j  that arise as the images of such a pre-node via the projections to U Y j , U Y j  as the branch cusps of the pre-node. Then we suppose further that every pre-node maps injectively into U X , and that the image in U X of the pre-node has empty intersec- tion with ∂U X , as well as with the image via ι j  , for j  ∈ J, of any cusp of U Y j  which is not a branch cusp of the pre- node. We shall refer to the image in U X of a pre-node as a node [of the HS-decomposition ({Y j } j∈J , {ι j } j∈J )]. Thus, [one verifies easily that] every node arises from a unique pre-node. We shall refer to the branch cusps of the pre- node that gives rise to a node as the branch cusps of the node. [Thus, by “Brouwer invariance of domain”, it fol- lows that, for any pre-node of U Y j × U X U Y j  , the maps ι j , ι j  determine a homeomorphism of the topological space ob- tained by gluing, along the associated node, suitable open neighborhoods of the branch cusps of U Y j , U Y j  onto the topological space constituted by a suitable open neighbor- hood of the associated node in U X .] (5) For any j ∈ J, every cusp of U Y j maps homeomorphically onto either a cusp of U X or a node of ({Y j } j∈J , {ι j } j∈J ) [cf. (4)]. Moreover, every cusp of U X arises in this way from a cusp of U Y j for some [necessarily uniquely determined] j ∈ J. [Thus, by “Brouwer invariance of domain” — together with a suitable gluing argument as in (4) — it follows that every cusp of U X admits an open neighborhood that COMBINATORIAL ANABELIAN TOPICS IV 51 arises, for some j ∈ J, as the homeomorphic image, via ι j , of a suitable open neighborhood of a cusp of U Y j .] If ({Y j }, {ι j }) is an HS-decomposition of X, then we shall re- fer to the triple (X, {Y j }, {ι j }) as a collection of HSD-data [where the “HSD” stands for “hyperbolic surface decomposi- tion”]. If X = (X, {Y j }, {ι j }) is a collection of HSD-data, then we shall refer to the topological space U X (respectively, [the closed subspace of U X corresponding to] an element of the [fi- nite] set {Y j }; a cusp of U X ; a node of ({Y j }, {ι j }) [cf. (4)]) as the underlying surface (respectively, a vertex; a cusp; a node) of X. Also, we shall refer to a cusp or node of X as an edge of X. Definition 2.3. Let X = (X, {Y j }, {ι j }) be a collection of HSD-data [cf. Definition 2.2, (iv)]. (i) We shall refer to the semi-graph G X defined as follows as the dual semi-graph of X: We take the set of vertices (respectively, open edges; closed edges) of G X is the [finite] set of vertices (respectively, cusps; nodes) of X [cf. Definition 2.2, (iv)]. For a vertex v and an edge e of X, we take the set of branches of e that abut to v to be the set of natural inclusions [i.e., that arise from X — cf. Definition 2.2, (iv)] from the edge of X corresponding to e into the topological space U Y j associated to the Y j corresponding to the vertex v. (ii) We shall refer to the connected semi-graph G X of temperoids [cf. Definition 2.1, (i)] defined as follows as the semi-graph of temperoids associated to X: We take the underly- ing semi-graph of G X to be G X [cf. (i)]. For each vertex v of G X , we take the connected temperoid of G X corresponding to v to be the connected temperoid determined by the category of topo- logical coverings with countably many connected components of the topological space U Y j [cf. Definition 2.2, (iii)] associated to the Y j corresponding to the vertex v. For each edge e of G X , we take the connected temperoid of G X corresponding to e to be the connected temperoid determined by the category of topological coverings with countably many connected compo- nents of the circle [cf. Definition 2.2, (i)] on U X corresponding to the edge e. For each branch b of G X , we take the morphism of temperoids corresponding to b to be the morphism obtained by pulling back topological coverings of the topological spaces under consideration. 52 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (iii) We shall say that a semi-graph of temperoids is of HSD-type if it is isomorphic to the semi-graph of temperoids associated to some collection of HSD-data [cf. (ii)]. Example 2.4 (Semi-graphs of temperoids of HSD-type asso- ciated to stable log curves). Let (g, r) be a pair of nonnegative def integers such that 2g − 2 + r > 0. Write S = Spec(C). In the fol- lowing, we shall apply the notation and terminology of the discussion entitled “Curves” in [CbTpI], §0. (i) Let S → (M g,r ) C be a C-valued point of (M g,r ) C . Write S log for the fs log scheme obtained by equipping S with the log log structure induced by the log structure of (M g,r ) C ; X log → S log for the stable log curve over S log corresponding to the resulting log strict (1-)morphism S log → (M g,r ) C ; d for the rank of the group-characteristic of S log [cf. [MT], Definition 5.1, (i)], i.e., log log → S an for the morphism of fs the number of nodes of X log ; X an log analytic spaces determined by the morphism X log → S log ; X an → S an for the underlying morphism of analytic spaces of log log log log → S an ; X an (C), S an (C) for the respective topological X an log spaces “X ” defined in [KN], (1.2), in the case where we take log log the “X” of [KN], (1.2), to be X an , S an , i.e., for T ∈ {X, S}, def log T an (C) = { (t, h) | t ∈ T an , h ∈ Hom gp (M T gp an ,t , S 1 ) such that h(f ) = f (t)/|f (t)| for every f ∈ O T × an ,t ⊆ M T gp an ,t } def — where we write S 1 = { u ∈ C | |u| = 1 } and M T an for the sheaf of monoids on T an that defines the log structure of log . Then, by considering the functoriality discussed in [KN], T an log log (1.2.5), and the respective maps X an (C) → X an , S an (C) → S an induced by the first projections, we obtain a commutative diagram of topological spaces and continuous maps log X an (C) −−−→ X an ⏐ ⏐ ⏐ ⏐   log (C) −−−→ S an . S an Now one verifies immediately from the various definitions in- log volved that S an (C) is homeomorphic to a product (S 1 ) ×d of d copies of S 1 ; moreover, it follows from [NO], Theorem 5.1, that the left-hand vertical arrow of the above diagram is a log (C). Thus, since [one ver- topological fiber bundle. Let s ∈ S an 1 ×d ifies easily that] (S ) is an Eilenberg-Maclane space [i.e., its COMBINATORIAL ANABELIAN TOPICS IV 53 universal covering space is contractible], the left-hand vertical arrow of the above diagram determines an exact sequence log log log 1 −→ π 1 (X an (C)| s ) −→ π 1 (X an (C)) −→ π 1 (S an (C)) ( ∼ = Z ⊕d ) −→ 1 log — where we write X an (C)| s for the fiber of the left-hand ver- log log tical arrow X an (C) → S an (C) of the above diagram at s — which thus determines an outer action log log π 1 (S an (C)) ( ∼ (C)| s )). = Z ⊕d ) −→ Out(π 1 (X an Write N ⊆ X an for the finite subset consisting of the nodes of log , C ⊆ X an for the finite subset consisting of the cusps of X an def log X an , U = X an \ (N ∪ C) ⊆ X an , and π 0 (U ) for the finite set of connected components of U . For each node x ∈ N (respectively, cusp y ∈ C; connected component F ∈ π 0 (U ) of U ), write C x log (respectively, C y ; Y F ) ⊆ X an (C)| s for the closure of the inverse image of {x} (respectively, {y}; F ) ⊆ X an via the composite pr log log X an (C)| s → 1 X an (C) → X an — where the second arrow is the upper horizontal arrow of the above diagram. Then one verifies immediately from the various definitions involved that there exists a uniquely determined, up to unique isomorphism [in the evident sense], collection of data as follows: • a pair of HS-type Z = (Z, {D i } ri=1 ) of type (g, r) [cf. Def- inition 2.2, (ii)]; ∼ log log • a homeomorphism φ : X an (C)| s → U Z of X an (C)| s with the interior U Z of Z [cf. Definition 2.2, (iii)] such that log φ restricts to a homeomorphism of y∈C C y ⊆ X an (C)| s r with ∂U Z = i=1 ∂D i ⊆ U Z [cf. Definition 2.2, (iii)]. Moreover, there exists a uniquely determined, up to unique isomorphism [in the evident sense], HS-decomposition of Z [cf. Definition 2.2, (iv)] such that the set of vertices (respectively, nodes; cusps) [cf. Definition 2.2, (iv)] of the resulting collec- tion of HSD-data [cf. Definition 2.2, (iv)] is {φ(Y F )} F ∈π 0 (U ) (respectively, {φ(C x )} x∈N ; {φ(C y )} y∈C ). We shall write G X log for the semi-graph of temperoids of HSD-type associated to this collection of HSD-data [cf. Definition 2.3, (ii)] and refer to G X log as the semi-graph of temperoids of HSD-type associated to X log . Then one verifies immediately from the functoriality discussed in [KN], (1.2.5), applied to the vertices, nodes, and cusps of the data under consideration, that the locally trivial log log fibration X an (C) → S an (C) determines an action log π 1 (S an (C)) ( ∼ = Z ⊕d ) −→ Aut(G X log ), 54 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI which is compatible, in the evident sense, with the outer action log log π 1 (S an (C)) −→ Out(π 1 (X an (C)| s )) discussed above. (ii) Let S log be the fs log scheme obtained by equipping S with the log structure given by the fs chart N 1 → 0 ∈ C and X log → S log a stable log curve of type (g, r) over S log [cf. [CmbGC], Example 2.5, in the case where k = C]. Then one verifies easily log that the classifying (1-)morphism S log → (M g,r ) C of X log → log S log factors as a composite S log → T log → (M g,r ) C — where the first arrow is a morphism that induces an isomorphism between the underlying schemes, and the second arrow is strict — and, moreover, if we write Y log → T log for the stable log log curve determined by the strict (1-)morphism T log → (M g,r ) C , then we have a natural isomorphism over S log ∼ X log −→ Y log × T log S log . We shall write def G X log = G Y log [cf. (i)] and refer to G X log as the semi-graph of temperoids of HSD-type associated to X log . Then, by pulling back the ac- tion of the second to last display of (i) via the homomor- log log phism π 1 (S an (C)) → π 1 (T an (C)) induced by the morphism log log S → T , we obtain an action π 1 (S log (C)) ( ∼ = Z) −→ Aut(G X log ), an together with a compatible outer action log log π 1 (S an (C)) −→ Out(π 1 (X an (C)| s )). Remark 2.4.1. One verifies easily that the discussion of Example 2.4, (ii), generalizes immediately to the case of arbitrary fs log schemes S log with underlying scheme S = Spec(C). Proposition 2.5 (Fundamental groups of semi-graphs of tem- peroids of HSD-type). Let G be a semi-graph of temperoids of HSD- type associated [cf. Definition 2.3, (ii), (iii)] to a collection of HSD-data X [cf. Definition 2.2, (iv)]. Write U X for the underlying surface of X [cf. Definition 2.2, (iv)] and B tp (U X ) for the connected temperoid [cf. [SemiAn], Definition 3.1, (ii)] deter- mined by the category of topological coverings with countably many con- nected components of the topological space U X . Then the following hold: COMBINATORIAL ANABELIAN TOPICS IV 55 (i) We have a natural equivalence of categories ∼ B tp (U X ) −→ B tp (G) [cf. Definition 2.1, (vi)]. In particular, B tp (G) is a connected temperoid. Write Π G for the tempered fundamental group [which is well-defined, up to inner automorphism] of the connected temperoid B tp (G) [cf. [SemiAn], Remark 3.2.1; the discussion of “Galois-countable temperoids” in [IUTeichI], Remark 2.5.3, (i)]. [Thus, the tem- pered group Π G admits a natural outer isomorphism with the topological fundamental group, equipped with the discrete topol- ogy, of the topological space U X .] We shall refer to this tem- pered group Π G as the fundamental group of G. (ii) Every connected finite étale covering H → G [cf. Definition 2.1, (vi)] admits a natural structure of semi-graph of temper- oids of HSD-type. (iii) The connected semi-graph of anabelioids G Σ [cf. Definition 2.1, (iii)] is of pro-Σ PSC-type [cf. [CmbGC], Definition 1.1, (i)]. Write Π G Σ for the [pro-Σ] fundamental group of G Σ . Then the natural morphism G → G Σ of semi-graphs of temperoids of Definition 2.1, (iii), induces a natural outer injection Π G → Π G Σ [cf. (i)]. Moreover, this natural outer injection determines an outer isomorphism ∼ Π G Σ −→ Π G Σ — where we write Π Σ G for the pro-Σ completion of Π G . (iv) Let z ∈ VCN(G) [cf. Definition 2.1, (v)]. Write Π G z for the tempered fundamental group [cf. [SemiAn], Remark 3.2.1] of the connected temperoid G z of G corresponding to z. Then the natural outer homomorphism Π G z −→ Π G is a Σ-compatible injection [cf. the discussion entitled “Groups” in §0]. (v) In the notation of (iii) and (iv), the closure of the image of the composite Π G z → Π G → Π G Σ of the outer injections of (iii) and (iv) is a VCN-subgroup of Π G Σ [cf. (iii); [CbTpI], Definition 2.1, (i)] associated to z ∈ VCN(G) = VCN(G Σ ). 56 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. A natural equivalence of categories as in assertion (i) may be obtained by observing that, after sorting through the various defini- tions involved, an object of B tp (U X ) [i.e., a topological covering of U X ] amounts to the same data as an object of B tp (G). Assertion (ii) follows immediately from the various definitions involved. Next, we verify assertion (iii). The assertion that G Σ is of pro-Σ PSC- type, as well as the assertion that the morphism G → G Σ determines an ∼ outer isomorphism Π Σ G → Π G Σ , follows immediately from the various definitions involved. Thus, the assertion that the morphism G → G Σ determines an outer injection Π G → Π G Σ follows from the well-known fact that the discrete group Π G injects into its pro-l completion for any l ∈ Primes [cf., e.g., [RZ], Proposition 3.3.15; [Prs], Theorem 1.7]. Next, we verify the injectivity portion of assertion (iv). Let us first observe that it follows immediately from the various definitions involved that the composite Π G z → Π G → Π G  [cf. Definition 2.1, (iii)] of the outer homomorphism under consideration and the outer injection of assertion (iii) [in the case where Σ = Primes] factors as the composite Π G z → Π G  z → Π G  of the outer homomorphism Π G z → Π G  z induced by the morphism G z → G  z of Definition 2.1, (ii), and the natural outer inclusion Π G  z → Π G  [cf. [SemiAn], Proposition 2.5, (i)]. Thus, to complete the verification of the injectivity portion of assertion (iv), it suffices to verify that the outer homomorphism Π G z → Π G  z is injective. On the other hand, this follows from the well-known fact that Π G z injects into its pro- l completion for any l ∈ Primes [cf., e.g., [RZ], Proposition 3.3.15; [Prs], Theorem 1.7]. This completes the proof of the injectivity portion of assertion (iv). Assertion (v) follows immediately from the various definitions involved. Finally, it follows immediately from assertions (iii) and (v), together with the evident pro-Σ analogue of [SemiAn], Proposition 2.5, (i), that the natural outer injection of assertion (iv) is Σ-compatible. This completes the proof of assertion (iv), hence also of Proposition 2.5.  Remark 2.5.1. In the notation of Proposition 2.5, as is discussed in Proposition 2.5, (i), the fundamental group Π G of the semi-graph of temperoids of HSD-type G is naturally isomorphic, up to inner auto- morphism, to the topological fundamental group, equipped with the discrete topology, of the compact orientable hyperbolic topological sur- face with compact boundary U X . In particular, Π G is finitely generated, torsion-free, and center-free and injects into its pro-l completion for COMBINATORIAL ANABELIAN TOPICS IV 57 any l ∈ Primes [cf. Proposition 2.5, (iii)]. Moreover, it holds that Cusp(G)  = ∅ [cf. Definition 2.1, (v)] if and only if Π G is free. Remark 2.5.2. In the situation of Example 2.4, (ii), write G X log for the Σ semi-graph of temperoids of HSD-type associated to X log ; G X log for the semi-graph of anabelioids of pro-Σ PSC-type of Proposition 2.5, (iii), in the case where we take the “G” of Proposition 2.5, (iii), to be G X log ; PSC-Σ for the semi-graph of anabelioids of pro-Σ PSC-type associated G X log log to X [cf. [CmbGC], Example 2.5]. Then it follows from Proposi- ∼ tion 2.5, (iii), that we have a natural outer isomorphism Π Σ G X log → Π G Σ log . On the other hand, by associating finite étale coverings of X log (C) to log étale coverings of Kummer type of X log [cf. [KN], Lemma X an log 2.2] and then restricting such finite étale coverings to X an (C)| s [cf. Ex- Σ . ample 2.4, (i)], we obtain an outer homomorphism Π G log → Π G PSC-Σ X X log Then one verifies immediately from the various definitions involved that the composite of the two outer homomorphisms ∼ Π G Σ log ←− Π Σ G log −→ Π G PSC-Σ log X X X is a graphic outer isomorphism [cf. [CmbGC], Definition 1.4, (i)], i.e., arises from a uniquely determined isomorphism of semi-graphs of an- abelioids ∼ Σ PSC-Σ G X . log −→ G X log Finally, one verifies easily that the above discussion generalizes im- mediately to the case of arbitrary fs log schemes S log with underlying scheme S = Spec(C) [cf. Remark 2.4.1]. Definition 2.6. Let G be a semi-graph of temperoids of HSD-type. Write Π G for the fundamental group of G. (i) Let z ∈ VCN(G) [cf. Definition 2.1, (v)]. Then we shall refer to a closed subgroup of Π G that belongs to the Π G -conjugacy class of closed subgroups determined by the image of the outer injection of the display of Proposition 2.5, (iv), as a VCN- subgroup of Π G associated to z ∈ VCN(G). If, moreover, z ∈ Vert(G) (respectively, ∈ Cusp(G); ∈ Node(G); ∈ Edge(G)) [cf. Definition 2.1, (v)], then we shall refer to a VCN-subgroup of Π G associated to z as a verticial (respectively, a cuspidal; a nodal; an edge-like) subgroup of Π G associated to z. (ii) Write G  → G for the universal covering of G correspond-  [cf. Definition 2.1, (v)]. Then ing to Π G . Let z  ∈ VCN( G) we shall refer to the VCN-subgroup Π z  ⊆ Π G [cf. (i)] deter-  as the VCN-subgroup of Π G associ- mined by z  ∈ VCN( G)  If, moreover, z  ∈ Vert( G)  (respectively, ated to z  ∈ VCN( G). 58 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI  ∈ Node( G);  ∈ Edge( G))  [cf. Definition 2.1, (v)], ∈ Cusp( G); then we shall refer to the VCN-subgroup of Π G associated to z  as the verticial (respectively, cuspidal; nodal; edge-like) sub- group of Π G associated to z  . (iii) Let (g, r) be a pair of nonnegative integers such that 2g−2+r > 0 and v ∈ Vert(G). Then we shall say that v is of type (g, r) if the “(g, r)” appearing in Definition 2.2, (ii), for the pair of HS-type corresponding to v coincides with (g, r). Thus, one verifies easily that v is of type (g, r) if and only if the number of the branches of edges of G that abut to v is equal to r, and, moreover, rank Z (Π ab v ) = 2g + max{0, r − 1} — where we use the notation Π v to denote a verticial subgroup associated to v. Remark 2.6.1. In the notation of Definition 2.6, it follows from Propo- sition 2.5, (iv), that every verticial subgroup of Π G is naturally isomor- phic, up to inner automorphism, to the topological fundamental group, equipped with the discrete topology, of a compact orientable hyperbolic topological surface with compact boundary. In particular, every verti- cial subgroup of Π G is finitely generated, torsion-free, and center-free and injects into its pro-l completion for any l ∈ Primes [cf. Proposi- tion 2.5, (iii)]. Moreover, it follows from Proposition 2.5, (iv), that every edge-like subgroup of Π G is naturally isomorphic, up to inner au- tomorphism, to the topological fundamental group, equipped with the discrete topology, of a unit circle [hence isomorphic to Z]. Definition 2.7. Let G and H be semi-graphs of temperoids of HSD- type. Write Π G , Π H for the fundamental groups of G, H, respectively. ∼ (i) We shall say that an isomorphism of groups Π G → Π H is group- theoretically verticial (respectively, group-theoretically cuspi- dal; group-theoretically nodal) if the isomorphism induces a bijection between the set of the verticial (respectively, cusp- idal; nodal) subgroups [cf. Definition 2.6, (i)] of Π G and the set of the verticial (respectively, cuspidal; nodal) subgroups of Π H . ∼ We shall say that an outer isomorphism Π G → Π H is group- theoretically verticial (respectively, group-theoretically cuspi- dal; group-theoretically nodal) if it arises from an isomorphism ∼ Π G → Π H that is group-theoretically verticial (respectively, group-theoretically cuspidal; group-theoretically nodal). ∼ (ii) We shall say that an outer isomorphism Π G → Π H is graphic ∼ if it arises from an isomorphism G → H. We shall say that COMBINATORIAL ANABELIAN TOPICS IV 59 ∼ an isomorphism Π G → Π H is graphic if the outer isomorphism ∼ Π G → Π H determined by it is graphic. Definition 2.8. Let G be a semi-graph of temperoids of HSD-type. Write G for the underlying semi-graph of G. Also, for each z ∈ VCN(G), write G z for the connected temperoid of G corresponding to z. (i) Let H be a sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of G. Then one may define a semi-graph of temperoids of HSD-type G| H as follows [cf. Fig. 2 of [CbTpI]]: We take the underlying semi- graph of G| H to be H; for each vertex v (respectively, edge e) of H, we take the temperoid corresponding to v (respectively, e) to be G v (respectively, G e ); for each branch b of an edge e of H that abuts to a vertex v of H, we take the morphism associated to b to be the morphism G e → G v associated to the branch of G corresponding to b. We shall refer to G| H as the semi-graph of temperoids of HSD-type obtained by restricting G to H. Thus, one has a natural morphism G| H −→ G of semi-graphs of temperoids. (ii) Let S ⊆ Cusp(G) be a subset of Cusp(G) [cf. Definition 2.1, (v)] which is omittable [cf. [CbTpI], Definition 2.4, (i)] as a subset  of the semi-graph of anabelioids of the set of cusps Cusp( G) of pro-Primes PSC-type G  [cf. Proposition 2.5, (iii), in the case where Σ = Primes] relative to the natural identification  Then, by eliminating the cusps contained Cusp(G) = Cusp( G). in S, and, for each vertex v of G, replacing the temperoid G v by the temperoid of coverings of G v that restrict to a trivial covering over the cusps contained in S that abut to v, we obtain a semi-graph of temperoids of HSD-type G •S [cf. Fig. 3 of [CbTpI]]. We shall refer to G •S as the partial compactification of G with respect to S. (iii) Let S ⊆ Node(G) be a subset of Node(G) [cf. Definition 2.1, (v)] such that the semi-graph obtained by removing the closed edges corresponding to the elements of S from the underly- ing semi-graph of G is connected, i.e., in the terminology of [CbTpI], Definition 2.5, (i), that is not of separating type as a  of the semi-graph of anabe- subset of the set of nodes Node( G) lioids of pro-Primes PSC-type G  [cf. Proposition 2.5, (iii), in 60 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI the case where Σ = Primes] relative to the natural identifica-  Then one may define a semi-graph tion Node(G) = Node( G). of temperoids of HSD-type G S as follows [cf. Fig. 4 of [CbTpI]]: We take the underlying semi- graph of G S to be the semi-graph obtained by replacing each node e of G contained in S such that V(e) = {v 1 , v 2 } ⊆ Vert(G) [cf. Definition 2.1, (v)] — where v 1 , v 2 are not necessarily dis- tinct — by two cusps that abut to v 1 , v 2 ∈ Vert(G), respec- tively, which we think as corresponding to the two branches of e. We take the temperoid corresponding to a vertex v (respectively, node e) of G S to be G v (respectively, G e ). [Note that the set of vertices (respectively, nodes) of G S may be naturally identified with Vert(G) (respectively, Node(G) \ S).] We take the temperoid corresponding to a cusp of G S arising from a cusp e of G to be G e . We take the temperoid corre- sponding to a cusp of G S arising from a node e of G to be G e . For each branch b of G S that abuts to a vertex v of a node e (respectively, of a cusp e that does not arise from a node of G), we take the morphism associated to b to be the mor- phism G e → G v associated to the branch of G corresponding to b. For each branch b of G S that abuts to a vertex v of a cusp of G S that arises from a node e of G, we take the mor- phism associated to b to be the morphism G e → G v associated to the branch of G corresponding to b. We shall refer to G S as the semi-graph of temperoids of HSD-type obtained from G by resolving S. Thus, one has a natural morphism G S −→ G of semi-graphs of temperoids. Remark 2.8.1. One verifies immediately that the operations of re- striction, partial compactification, and resolution discussed in Defini- tion 2.8, (i), (ii), (iii), are compatible [in the evident sense] with the corresponding pro-Σ operations — i.e., as discussed in [CbTpI], Defini- tion 2.2, (ii); [CbTpI], Definition 2.4, (ii); [CbTpI], Definition 2.5, (ii) — relative to the operation of passing to the associated semi-graph of anabelioids of pro-Σ PSC-type [cf. Proposition 2.5, (iii)]. Remark 2.8.2. We take this opportunity to correct an unfortunate misprint in [CbTpI], Definition 2.5, (ii): the phrase “by two cusps that abut to v 1 , v 2 ∈ Vert(G), respectively” of [CbTpI], Definition 2.5, (ii), COMBINATORIAL ANABELIAN TOPICS IV 61 should read “by two cusps that abut to v 1 , v 2 ∈ Vert(G), respectively, which we think as corresponding to the two branches of e”. Definition 2.9. In the notation of Definition 2.8, let S ⊆ Node(G) be a subset of Node(G) [cf. Definition 2.1, (v)]. Then we define the semi-graph of temperoids of HSD-type G S as follows [cf. Fig. 5 of [CbTpI]]: def (i) We take Cusp(G S ) = Cusp(G) [cf. Definition 2.1, (v)]. def (ii) We take Node(G S ) = Node(G) \ S [cf. Definition 2.1, (v)]. (iii) We take Vert(G S ) [cf. Definition 2.1, (v)] to be the set of connected components of the semi-graph obtained from G by omitting the edges e ∈ Edge(G) \ S [cf. Definition 2.1, (v)]. Alternatively, one may take Vert(G S ) to be the set of equiva- lence classes of elements of Vert(G) with respect to the equiva- lence relation “∼” defined as follows: for v, w ∈ Vert(G), v ∼ w if either v = w or there exist n elements e 1 , . . . , e n ∈ S of S and def n + 1 vertices v 0 , v 1 , . . . , v n ∈ Vert(G) of G such that v 0 = v, def v n = w, and, for 1 ≤ i ≤ n, it holds that V(e i ) = {v i−1 , v i } [cf. Definition 2.1, (v)]. (iv) For each branch b of an edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) and each vertex v ∈ Vert(G S ) of G S , b abuts, relative to G S , to v if b abuts, relative to G, to an element of the equivalence class v [cf. (iii)]. (v) For each edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) of G S , we take the temperoid of G S corresponding to e ∈ Edge(G S ) to be the temperoid G e . (vi) Let v ∈ Vert(G S ) be a vertex of G S . Then one verifies easily that there exists a unique sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] H v of the underlying semi- graph of G whose set of vertices consists of the elements of the equivalence class v [cf. (iii)]. Write def T v = Node(G| H v ) \ (S ∩ Node(G| H v )) [cf. Definition 2.8, (i)]. Then we take the temperoid of G S cor- responding to v ∈ Vert(G S ) to be the temperoid B tp ((G| H v ) T v ) [cf. Definition 2.1, (vi); Proposition 2.5, (i); Definition 2.8, (iii)]. (vii) Let b be a branch of an edge e ∈ Edge(G S ) (= Edge(G) \ S — cf. (i), (ii)) that abuts to a vertex v ∈ Vert(G S ). Then since b abuts to v, one verifies easily that there exists a unique vertex w of G which belongs to the equivalence class v [cf. (iii)] such that 62 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI b abuts to w relative to G. We take the morphism of temperoids associated to b, relative to G S , to be the morphism naturally determined by post-composing the morphism of temperoids G e → G w corresponding to the branch b relative to G with the natural morphism of temperoids G w → B tp ((G| H v ) T v ) [cf. (vi)]. We shall refer to this semi-graph of temperoids of HSD-type G S as the generization of G with respect to S. Remark 2.9.1. One verifies immediately that the operation of gener- ization discussed in Definition 2.9 is compatible [in the evident sense] with the corresponding pro-Σ operation — i.e., as discussed in [CbTpI], Definition 2.8 — relative to the operation of passing to the associated semi-graph of anabelioids of pro-Σ PSC-type [cf. Proposition 2.5, (iii)]. Remark 2.9.2. We take this opportunity to correct an unfortunate misprint in [CbTpI], Definition 2.8, (vii): the phrase “equivalent class” should read “equivalence class”. Proposition 2.10 (Specialization outer isomorphisms). Let G be a semi-graph of temperoids of HSD-type and S ⊆ Node(G) a subset of Node(G). Write Π G for the fundamental group of G and Π G S for the fundamental group of the generization G S of G with respect to S [cf. Definition 2.9]. Then there exists a natural outer isomorphism ∼ Φ G S : Π G S −→ Π G which is functorial, in the evident sense, with respect to isomorphisms of the pair (G, S) and satisfies the following three conditions: (a) Φ G S induces a bijection between the set of cuspidal subgroups [cf. Definition 2.6, (i)] of Π G S and the set of cuspidal sub- groups of Π G . (b) Φ G S induces a bijection between the set of nodal subgroups [cf. Definition 2.6, (i)] of Π G S and the set of nodal subgroups of Π G associated to the elements of Node(G) \ S. (c) Let v ∈ Vert(G S ) be a vertex of G S ; H v , T v as in Defini- tion 2.9, (vi). Then Φ G S induces a bijection between the Π G S - conjugacy class of any verticial subgroup [cf. Definition 2.6, (i)] Π v ⊆ Π G S of Π G S associated to v ∈ Vert(G S ) and the Π G - conjugacy class of subgroups obtained by forming the image of the outer homomorphism Π (G| H v ) Tv −→ Π G induced by the natural morphism (G| H v ) T v → G [cf. Defini- tion 2.8, (i), (iii)] of semi-graphs of temperoids. COMBINATORIAL ANABELIAN TOPICS IV 63 We shall refer to this natural outer isomorphism Φ G S as the spe- cialization outer isomorphism with respect to S. Proof. An outer isomorphism that satisfies the three conditions in the statement of Proposition 2.10 may be obtained by observing that, after sorting through the various definitions involved, an object of B tp (G S ) amounts to the same data as an object of B tp (G). This completes the proof of Proposition 2.10.  Remark 2.10.1. One verifies immediately that the specialization outer isomorphism discussed in Proposition 2.10 is compatible [in the evident sense] with the corresponding pro-Σ outer isomorphism — i.e., as dis- cussed in [CbTpI], Proposition 2.9 — relative to the operation of pass- ing to the associated semi-graph of anabelioids of pro-Σ PSC-type [cf. Proposition 2.5, (iii)]. Lemma 2.11 (Infinite cyclic coverings). Let G be a semi-graph of temperoids of HSD-type. Suppose that (Vert(G)  , Node(G)  ) = (1, 1), i.e., the semi-graph of anabelioids of pro-Primes PSC-type G  [cf. Propo- sition 2.5, (iii), in the case where Σ = Primes] is cyclically primitive [cf. [CbTpI], Definition 4.1]. Write Π G for the fundamental group of G; G for the underlying semi-graph of G; Π G ( ∼ = Z) for the discrete topological fundamental group of G; G ∞ → G for the connected cover- ing of G [cf. Definition 2.1, (vi)] corresponding to the natural surjection def Π G  Π G ; Π G ∞ = Ker(Π G  Π G ). Then the following hold: ∼ (i) Fix an isomorphism Π G → Z. Then there exists a triple of bijections ∼ ∼ V : Z −→ Vert(G ∞ ), N : Z −→ Node(G ∞ ), ∼ C : Z × Cusp(G) −→ Cusp(G ∞ ) [cf. Definition 2.1, (v)] that satisfies the following properties: • The bijections are equivariant with respect to the action ∼ of Π G → Z on Z by translations and the natural action of Π G on “Vert(−)”, “Node(−)”, “Cusp(−)”. • The post-composite of C with the natural map Cusp(G ∞ ) → Cusp(G) coincides with the projection Z × Cusp(G) → Cusp(G) to the second factor. • For each a ∈ Z, it holds that E(V (a)) = {N (a), N (a + 1)}  { C(a, z) | z ∈ Cusp(G) } [cf. Definition 2.1, (v)]. Moreover, such a triple of bijections is unique, up to post- composition with the automorphisms of “Vert(−)”, “Node(−)”, “Cusp(−)” determined by the action of a [single!] element of Π G . 64 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (ii) Let a ≤ b be integers. Write G [a,b] for the [uniquely deter- mined] sub-semi-graph of PSC-type [cf. [CbTpI], Definition 2.2, (i)] of the underlying semi-graph of G ∞ whose set of ver- tices is equal to {V (a), V (a+1), . . . , V (b)} [cf. (i)]. Also, write G [a,b] for the semi-graph of temperoids obtained by restricting G ∞ to G [a,b] [in the evident sense — cf. also the procedure dis- cussed in Definition 2.8, (i)]. Then G [a,b] is a semi-graph of temperoids of HSD-type. (iii) Let a ≤ b be integers. For an integer c such that a ≤ c ≤ b (respectively, a + 1 ≤ c ≤ b), let Π V (c) ⊆ Π G [a,b] (respectively, Π N (c) ⊆ Π G [a,b] ) be a verticial (respectively, nodal) subgroup of Π G [a,b] associated to V (c) ∈ Vert(G [a,b] ) (respectively, N (c) ∈ Node(G [a,b] )) [cf. (i), (ii)] such that, for a + 1 ≤ c ≤ b, it holds that Π N (c) ⊆ Π V (c−1) ∩ Π V (c) . Then the inclusions Π V (c) , Π N (c) → Π G [a,b] determine an isomorphism   lim Π V (a) ← Π N (a+1) → Π V (a+1) ← · · · → Π V (b−1) ← Π N (b) → Π V (b) −→ ∼ −→ Π G [a,b] — where lim denotes the inductive limit in the category of −→ groups. (iv) Let a ≤ b be integers. Then the composite G [a,b] → G ∞ → G determines an outer injection Π G [a,b] → Π G . Moreover, the image of this outer injection is contained in the normal subgroup Π G ∞ ⊆ Π G . (v) There exists a collection {D [−a,a] } 1≤a∈Z of subgroups D [−a,a] ⊆ Π G ∞ indexed by the positive integers which satisfy the following properties: • D [−a,a] ⊆ Π G ∞ belongs to the Π G -conjugacy class [of sub- groups of Π G ] obtained by forming the image of the outer injection Π G [−a,a] → Π G of (iv). • D [−a,a] ⊆ D [−a−1,a+1] . • The inclusions D [−a,a] → Π G [where a ranges over the positive integers] determine an isomorphism   ∼ lim D [−1,1] → D [−2,2] → D [−3,3] → · · · −→ Π G ∞ −→ — where lim denotes the inductive limit in the category of −→ groups. (vi) In the situation of (v), since Π G injects into its pro-l com- pletion for any l ∈ Primes [cf. Remark 2.5.1], let us regard subgroups of Π G as subgroups of the pro-Σ completion Π Σ G of Σ Π G . Let a be a positive integer. Write D [−a,a] ⊆ Π G for the closure of D [−a,a] in Π Σ  ∈ Π Σ G . Let γ G . Suppose that D [a,−a] ∩ COMBINATORIAL ANABELIAN TOPICS IV 65 γ  · D [a,−a] · γ  −1  = {1}. Then the image of γ  ∈ Π Σ G in the pro-Σ Σ Σ completion Π G of Π G is contained in Π G ⊆ Π G . (vii) In the situation of (vi), suppose, moreover, that γ  is contained Σ of Π in Π . Then γ  ∈ D [a,−a] . in the closure Π G ∞ ⊆ Π Σ G ∞ G G Proof. Assertions (i), (ii) follow immediately from the various defini- tions involved. Assertion (iii) follows immediately from a similar argu- ment to the argument applied in the proof of [CmbCsp], Proposition 1.5, (iii). Next, we verify assertion (iv). The injectivity portion of asser- tion (iv) follows immediately — by considering a suitable finite étale subcovering of G ∞ → G and applying a suitable specialization outer isomorphism [cf. Proposition 2.10] — from Proposition 2.5, (iv). The remainder of assertion (iv) follows immediately from the various defi- nitions involved. This completes the proof of assertion (iv). Assertion (v) follows immediately from assertion (iii). Next, we verify assertion (vi). Write G Σ for the semi-graph of an- abelioids of pro-Σ PSC-type determined by G [cf. Proposition 2.5, (iii)], G  Σ → G Σ for the universal covering of the semi-graph of anabe- lioids of pro-Σ PSC-type G corresponding to [the torsion-free group]  Σ for the Π Σ G [cf. Proposition 2.5, (iii); [MT], Remark 1.2.2], and G underlying pro-semi-graph of G  Σ . Then it follows immediately — i.e., by considering a suitable finite étale subcovering of G ∞ → G and applying a suitable specialization outer isomorphism [cf. Propo- sition 2.10] — from [NodNon], Lemma 1.9, (ii), that our assumption that D [a,−a] ∩ γ  · D [a,−a] · γ  −1  = {1} implies that the respective sub-  Σ determined by D [a,−a] , γ pro-semi-graphs of G  · D [a,−a] · γ  −1 ⊆ Π Σ G [cf. Proposition 2.5, (v)] either contain a common pro-vertex or may be joined to one another by a single pro-edge. But this implies that γ  maps G [−a,a] to some Π G -translate of G [−a,a] , hence, in particular, Σ Σ that the image of γ  ∈ Π Σ G in Π G is contained in Π G ⊆ Π G , as desired. This completes the proof of assertion (vi). Assertion (vii) follows im- mediately — i.e., by considering a suitable finite étale subcovering of G ∞ → G and applying a suitable specialization outer isomorphism [cf. Proposition 2.10] — from the commensurable terminality [cf. [CmbGC], Proposition 1.2, (ii)] of D [a,−a] in a suitable open subgroup of Π Σ G con- taining Π G ∞ [cf. also [NodNon], Lemma 1.9, (ii)]. This completes the proof of Lemma 2.11.  The content of the following lemma is entirely elementary and well- known. Lemma 2.12 (Action of the symplectic group). Let g be a pos- itive integer. For each positive integer n and v = (v 1 , . . . , v n ) ∈ Z ⊕n , write vol(v) ∈ Z for the [uniquely determined] nonnegative integer that 66 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI generates the ideal Z · v 1 + · · · + Z · v n ⊆ Z; M n (Z) for the set of n by n matrices with coefficients in Z; GL n (Z) ⊆ M n (Z) for the group of matrices A ∈ M n (Z) such that det(A) ∈ {1, −1}; Sp 2g (Z) ⊆ GL 2g (Z) for the subgroup of 2g by 2g symplectic matrices, i.e., B ∈ GL 2g (Z) such that   0 1 t 0 1 B · · B = . −1 0 −1 0 [Note that one verifies immediately that, for every A ∈ GL n (Z), it holds that vol(v) = vol(vA).] Then the following hold: (i) Let v = (v 1 , . . . , v g ) ∈ Z ⊕g . Then there exists an invertible g−1    matrix A ∈ GL g (Z) such that vA = (vol(v), 0, . . . , 0). (ii) Let v = (v 1 , . . . , v 2g ) ∈ Z ⊕2g . Then there exists a symplectic 2g−1    matrix B ∈ Sp 2g (Z) such that vB = (vol(v), 0, . . . , 0). (iii) Let N ⊆ Z ⊕2g be a submodule of Z ⊕2g and v ∈ Z ⊕2g . Suppose that N  = {0}. Then there exist a nonzero integer n ∈ Z \ {0} and a symplectic matrix B ∈ Sp 2g (Z) such that n · vB ∈ N . (iv) Let N ⊆ Z ⊕2g be a submodule of Z ⊕2g and π : Z ⊕2g  Z a surjection. Suppose that N is of infinite index in Z ⊕2g . Then there exists a symplectic matrix B ∈ Sp 2g (Z) such that N · B ⊆ Ker(π). Proof. First, we verify assertion (i). Let us first observe that if v = 0 [i.e., vol(v) = 0], then assertion (i) is immediate. Thus, to verify assertion (i), we may assume without loss of generality that v  = 0. In particular, to verify assertion (i), by replacing v by vol(v) −1 · v ∈ Z ⊕g , we may assume without loss of generality that vol(v) = 1. On the other hand, since vol(v) = 1, one verifies immediately that Z ⊕g /(Z · v) is a free Z-module of rank g − 1, hence that there exists an injection ∼ Z ⊕g−1 → Z ⊕g that induces an isomorphism (Z · v) ⊕ Z ⊕g−1 → Z ⊕g . This completes the proof of assertion (i). Next, we verify assertion (ii). Since [one verifies easily that] Sp 2 (Z) = SL 2 (Z) = { B ∈ GL 2 (Z) | det(B) = 1 }, assertion (ii) in the case where g = 1 follows immediately from assertion (i) [in the case where we take “g” in assertion (i) to be 2], together with the [easily verified] fact that     a b a −b  a b det , det = {1, −1} for every ∈ GL 2 (Z). c d c −d c d For i ∈ {1, . . . , g}, write M i for the submodule of Z ⊕2g generated by (0, . . . , 0, 1, 0, . . . , 0), (0, . . . , 0, 1, 0, . . . , 0) ∈ Z ⊕2g — where the “1’s” lie, respectively, in the i-th and (g + i)-th compo- nents. Then, by applying assertion (ii) in the case where g = 1 [already verified above] to the M i ’s, we conclude that, to complete the verifi- cation of assertion (ii), we may assume without loss of generality that COMBINATORIAL ANABELIAN TOPICS IV 67 def v i = 0 for every g + 1 ≤ i ≤ 2g. Write v ≤g = (v 1 , . . . , v g ) ∈ Z ⊕g . Then let us observe that it follows from assertion (i) that there exists an invertible matrix A ∈ GL g (Z) such that v ≤g A = (vol(v ≤g ), 0, . . . , 0) = (vol(v), 0, . . . , 0). Thus, assertion (ii) follows immediately from the [easily verified] fact that  A 0 ∈ Sp 2g (Z). 0 t A −1 This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii). Assertion (iv) ⊕2g follows immediately — by applying the self-duality with respect  of Z 0 1 to the symplectic form determined by — from assertion (iii). −1 0 This completes the proof of Lemma 2.12.  Lemma 2.13 (Automorphisms of surface groups). Let g be a positive integer, Π the topological fundamental group of a connected orientable compact topological surface of genus g, π : Π  Z a surjec- tion, and J ⊆ Π a subgroup of Π such that the image of J in Π ab is of infinite index in Π ab . [For example, this will be the case if J is generated by 2g − 1 elements.] Then there exists an automorphism σ of Π such that σ(J) ⊆ Ker(π). def Proof. Write H = Hom(Π, Z) = Hom Z (Π ab , Z). Let us fix isomor- ∼ ∼ phisms H → Z ⊕2g and H 2 (Π, Z) → Z. Then it follows from the well-known theory of Poincaré duality that the cup product in group cohomology H × H = H 1 (Π, Z) × H 1 (Π, Z) −→ H 2 (Π, Z) ∼ = Z determines a perfect pairing on H; moreover, if we write Aut PD (H) ⊆ ∼ Aut(H) ( → GL 2g (Z) — cf. the notation of Lemma 2.12) for the sub- group of automorphisms of H that are compatible with this perfect ∼ pairing, then — by replacing the isomorphism H → Z ⊕2g by a suit- ∼ able isomorphism if necessary — the isomorphism Aut(H) → GL 2g (Z) ∼ determines an isomorphism Aut PD (H) → Sp 2g (Z) [cf. the notation of Lemma 2.12]. On the other hand, recall [cf., e.g., the discussion preced- ing [DM], Theorem 5.13] that the natural homomorphism Aut(Π) → Aut(H) determines a surjection Aut(Π)  Aut PD (H) (⊆ Aut(H)). Thus, Lemma 2.13 follows immediately from Lemma 2.12, (iv). This completes the proof of Lemma 2.13.  Lemma 2.14 (Finitely generated subgroups of surface groups). Let G be a semi-graph of temperoids of HSD-type and J ⊆ Π G a finitely 68 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI generated subgroup of the fundamental group Π G of G. Then the fol- lowing hold: (i) Suppose that Cusp(G)  = ∅. Then there exist a subgroup F ⊆ Π G of finite index and a surjection F  J such that J ⊆ F , and, moreover, the restriction of the surjection F  J to J ⊆ F is the identity automorphism of J. (ii) Suppose that (Vert(G)  , Cusp(G)  , Node(G)  ) = (1, 0, 1). Thus, since we are in the situation of Lemma 2.11, we shall apply the notational conventions established in Lemma 2.11. Suppose ab that the image of J in Π ab G is of infinite index in Π G . [For example, this will be the case if J is generated by rank Z (Π ab G )−1 elements.] Then there exists an automorphism σ ∈ Aut(Π G ) of Π G such that σ(J) ⊆ Π G ∞ . (iii) In the situation of (ii), suppose, moreover, that J ⊆ Π G ∞ . Then there exists a positive integer a ∈ Z such that J ⊆ D [−a,a] [cf. Lemma 2.11, (v)]. Proof. Assertion (i) follows from [SemiAn], Corollary 1.6, (ii), together with the fact that Π G is a finitely generated free group [cf. Remark 2.5.1]. Assertion (ii) follows from Lemma 2.13. Assertion (iii) follows from Lemma 2.11, (v), together with our assumption that J is finitely gen- erated. This completes the proof of Lemma 2.14.  Theorem 2.15 (Profinite conjugates of finitely generated Primes- compatible subgroups). Let Π be the topological fundamental group of a compact orientable hyperbolic topological surface with compact bound- ary [cf. Remark 2.5.1] and H, J ⊆ Π subgroups. Since Π injects into its pro-l completion for any l ∈ Primes [cf., e.g., [RZ], Proposition 3.3.15; [Prs], Theorem 1.7], let us regard subgroups of Π as subgroups  of Π. Write H, J ⊆ Π  for the closures of the profinite completion Π  respectively. Suppose that the following conditions are of H, J in Π, satisfied: (a) The subgroups H and J are finitely generated. (b) If J is of infinite index in Π, then J is of infinite index in  Π. [Here, we note that condition (b) is automatically satisfied whenever Π is free — cf. [SemiAn], Corollary 1.6, (ii).] Then the following hold: (i) It holds that J = J ∩ Π.  such that (ii) Suppose that there exists an element γ  ∈ Π H ⊆ γ  · J · γ  −1 . Then there exists an element δ ∈ Π such that H ⊆ δ · J · δ −1 . COMBINATORIAL ANABELIAN TOPICS IV 69 Proof. Let us first observe that, to verify Theorem 2.15, we may assume without loss of generality that Π is the fundamental group Π G of a semi-graph of temperoids of HSD-type G [cf. Definition 2.3]. Next, we claim that the following assertion holds: Claim 2.15.A: Theorem 2.15 holds in the case where J is of finite index in Π G . Indeed, write N ⊆ Π G for the normal subgroup of Π G obtained by forming the intersection of all Π G -conjugates of J. Then since J is of finite index in Π G , it is immediate that N is of finite index in Π G . Thus, by considering the images in Π G /N of the various groups involved, one verifies immediately that Theorem 2.15 holds in the case where J is of finite index in Π G . This completes the proof of Claim 2.15.A. Thus, in the remainder of the proof of Theorem 2.15, we may assume without loss of generality that J is of infinite index in Π G , which implies that  G [cf. condition (b)]. J is of infinite index in Π Next, we claim that the following assertion holds: Claim 2.15.B: Let F ⊆ Π G be a subgroup of finite index such that J ⊆ F . Suppose that the assertion obtained by replacing Π G in assertion (i) by F holds. Then assertion (i) holds, and, in the situation of as- sertion (ii), there exists a Π G -conjugate of H that is contained in F . If, moreover, the assertion obtained by replacing Π G in assertion (ii) by F holds, then as- sertion (ii) holds. Indeed, let us first observe that since the natural inclusion F → Π G is Primes-compatible [cf. the discussion entitled “Groups” in §0], the profinite completion F  of F may be identified with the closure F of F  G . In particular, the closure of J in F  is naturally isomorphic to the in Π  G . Thus, it follows from Claim 2.15.A applied to F closure J of J in Π that the assertion obtained by replacing Π G in assertion (i) by F implies assertion (i). Next, let us observe that in the situation of assertion (ii),  G , by replacing H by a since [one verifies immediately that] Π G · F = Π suitable Π G -conjugate of H, we may assume without loss of generality that γ  ∈ F . In particular, since H ⊆ γ  · J · γ  −1 ⊆ γ  · F · γ  −1 = F , it follows that H ⊆ F ∩ Π G = F [cf. Claim 2.15.A]. Thus, one verifies easily that the assertion obtained by replacing Π G in assertion (ii) by F implies assertion (ii). This completes the proof of Claim 2.15.B. Next, we verify Theorem 2.15 in the case where Cusp(G)  = ∅. Suppose that Cusp(G)  = ∅. Then it follows from Lemma 2.14, (i), that there exist a subgroup F ⊆ Π G of finite index and a surjection π : F  J such that J ⊆ F , and, moreover, the restriction of π to J ⊆ F is the identity automorphism of J. Now it follows immediately from Claim 2.15.B that, by replacing Π G by F , we may assume without loss of generality that Π G = F . Next, let us observe that since [it is 70 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI immediate that] J ⊆ J ∩ Π G , to complete the verification of assertion (i) in the case where Cusp(G)  = ∅, it suffices to verify that J ∩ Π G ⊆ J. Moreover, since J ⊆ J ∩ Π G (⊆ J), it follows immediately from the  G  J for the surjection in- equality π  | J = id J [where we write π  : Π duced by π] that, to verify the inclusion J ∩ Π G ⊆ J, it suffices to verify that π  (J ∩ Π G ) ⊆ π  (J). On the other hand, one verifies easily that π  (J ∩ Π G ) ⊆ π  (Π G ) = J = π  (J), as desired. This completes the proof of assertion (i) in the case where Cusp(G)  = ∅. Next, to verify assertion (ii) in the case where Cusp(G)  = ∅, let us observe that, by replacing γ  by γ  · π  ( γ −1 ), we may assume without loss of generality that γ  ∈ Ker( π ). Now we claim that the following assertion holds: Claim 2.15.C: It holds that H ⊆ γ  · J · γ  −1 .  , J ⊆ J, it follows Indeed, since [one verifies easily that] γ  −1 · H · γ immediately from the equality π  | J = id J that, to verify Claim 2.15.C, it suffices to verify that π  ( γ −1 · H · γ  ) ⊆ π  (J). On the other hand, since γ  ∈ Ker( π ), it holds that  ) = π  (H) ⊆ π  (Π G ) = J = π  (J), π  ( γ −1 · H · γ as desired. This completes the proof of Claim 2.15.C. In particular, it follows immediately from [IUTeichI], Theorem 2.6 [i.e., in essence, the argument given in the proof of [André], Lemma 3.2.1], that there  −1 · H · γ  ⊆ J. This exists an element δ ∈ Π G such that δ −1 · H · δ = γ completes the proof of assertion (ii) in the case where Cusp(G)  = ∅, hence also of Theorem 2.15 in the case where Cusp(G)  = ∅. Next, we verify Theorem 2.15 in the case where Cusp(G) = ∅. Sup- pose that Cusp(G) = ∅. First, we observe that since J is of infinite  G , it follows immediately that [Π G : J ·N ] → +∞ as N ranges index in Π over the normal subgroups of Π G of finite index, hence [cf. Claim 2.15.B; the fact that J is finitely generated] that, by replacing Π G by a suitable subgroup of finite index in Π G that contains J, we may assume without ab loss of generality that the image of J in Π ab G is of infinite index in Π G [cf. Remark 2.5.1]. Moreover, by considering suitable specialization outer isomorphisms [cf. Proposition 2.10], we may assume without loss of generality that the equality (Vert(G)  , Cusp(G)  , Node(G)  ) = (1, 0, 1) holds. Thus, since we are in the situation of Lemma 2.11, we shall apply the notational conventions established in Lemma 2.11. More- over, it follows from Lemma 2.14, (ii), that, by considering a suitable automorphism of Π G , we may assume without loss of generality that J ⊆ Π G ∞ . Thus, it follows from Lemma 2.14, (iii), that there exists a positive integer a ∈ Z such that J ⊆ D [−a,a] ⊆ Π G ∞ . COMBINATORIAL ANABELIAN TOPICS IV 71 ∼ Next, let us observe that since Π G /Π G ∞ → Π G ( ∼ = Z) injects into its profinite completion, it follows that J ∩ Π G ⊆ Π G ∞ . In particular, by applying Lemma 2.14, (iii), we conclude that, for any given fixed element α ∈ J ∩ Π G , we may assume, by possibly enlarging a, that α ∈ D [−a,a] . Next, let us observe — i.e., by considering a suitable finite étale subcovering of G ∞ → G and applying a suitable specialization outer isomorphism [cf. Proposition 2.10] — that the natural inclusion D [−a,a] → Π G is Primes-compatible [cf. Proposition 2.5, (iv)]. In par- ticular, by replacing G by G [−a,a] [cf. Lemma 2.11, (ii)], we conclude that assertion (i) in the case where Cusp(G) = ∅ follows from asser- tion (i) in the case where Cusp(G)  = ∅ [already verified above]. This completes the proof of assertion (i) in the case where Cusp(G) = ∅. Finally, to verify assertion (ii) in the case where Cusp(G) = ∅, let us observe that if H = {1}, then assertion (ii) is immediate. Thus, we may assume without loss of generality that H  = {1}. Next, let us observe ∼ that since J ⊆ D [−a,a] ⊆ Π G ∞ , and Π G /Π G ∞ → Π G ( ∼ = Z) injects into its profinite completion, one verifies immediately that H ⊆ Π G ∞ . Thus, since H ⊆ Π G ∞ is finitely generated, it follows from Lemma 2.14, (iii), that, by possibly enlarging a, we may assume without loss of generality  · J · γ  −1 ⊆ that H ⊆ D [−a,a] . Since, moreover, {1}  = H ⊆ D [−a,a] ∩ γ D [−a,a] ∩ γ ·D [−a,a] · γ −1 , it follows from Lemma 2.11, (vi), that the image  G of Π G is contained in Π G ⊆ Π  G ,  G in the profinite completion Π of γ  ∈ Π  which thus implies that there exists an element γ ∈ Π G such that γ  γ  ∈ Π G ∞ . In particular, by replacing H by γ  · H · (γ  ) −1 and possibly enlarging a, we may assume without loss of generality that γ  ∈ Π G ∞ . Thus, again by applying the fact that {1}  = D [−a,a] ∩ γ  · D [−a,a] · γ  −1 , we conclude from Lemma 2.11, (vii), that γ  ∈ D [−a,a] . In particular, since, as discussed above [cf. the discussion immediately preceding the proof of assertion (i) in the case where Cusp(G) = ∅], the natural inclusion D [−a,a] → Π G is Primes-compatible, by replacing G by G [−a,a] , we conclude that assertion (ii) in the case where Cusp(G) = ∅ follows from assertion (ii) in the case where Cusp(G)  = ∅ [already verified above]. This completes the proof of assertion (ii) in the case where Cusp(G) = ∅, hence also of Theorem 2.15.  Remark 2.15.1. In passing, we observe that the analogue of Theo- rem 2.15 for arbitrary Σ  = Primes is false. Indeed, if, in the statement of Theorem 2.15, one replaces “Π” by the group Z, then it is easy to construct counterexamples to assertions (i), (ii). One may then obtain counterexamples in the case of the original “Π” by considering the case where the original “Π” is the fundamental group Π G of a semi-graph of temperoids of HSD-type G such that Edge(G)  = ∅ and considering suitable edge-like subgroups [i.e., isomorphic to Z!] of Π G . 72 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Lemma 2.16 (VCN-subgroups of infinite index). Let G be a semi- graph of anabelioids of pro-Σ PSC-type (respectively, of temperoids of def def HSD-type). Write J = Π Σ G (respectively, J = Π G ) for the [pro-Σ (respectively, discrete)] fundamental group of G. Let H ⊆ J be a VCN- subgroup of J. Consider the following two [mutually exclusive] condi- tions: (1) H = J. (2) H is of infinite index in J. Then we have equivalences (1) ⇐⇒ (1  ); (2) ⇐⇒ (2  ) with the following two conditions: (1  ) H is verticial, and Node(G) = ∅. (2  ) Either H is edge-like, or Node(G)  = ∅. Proof. The implication (1  ) ⇒ (1) follows immediately from the various definitions involved. Thus, one verifies immediately [by considering suitable contrapositive versions of the various implications involved] that, to complete the verification of Lemma 2.16, it suffices to verify the implication (2  ) ⇒ (2). To this end, let us observe that if H is edge- like, then since H is abelian, and every closed subgroup of J of finite index is center-free [cf., e.g., Remark 2.5.1; [CmbGC], Remark 1.1.3], we conclude that H is of infinite index in J. Thus, we may assume without loss of generality that H is verticial and Node(G)  = ∅. Now since Node(G)  = ∅, it follows from a similar argument to the argument in the discussion entitled “Curves” in [AbsTpII], §0, that, by replacing G by a suitable connected finite étale covering of G, we may assume without loss of generality that the underlying semi-graph of G is loop- ample [cf. the discussion entitled “Semi-graphs” in [AbsTpII], §0]. In particular, since [one verifies easily that] the abelianization of the [pro- Σ completion of the] topological fundamental group of a noncontractible semi-graph is infinite, the image of H in the abelianization of J is of infinite index, which thus implies that H is of infinite index in J, as desired. This completes the proof of Lemma 2.16.  Corollary 2.17 (Profinite conjugates of VCN-subgroups). Let G and H be semi-graphs of temperoids of HSD-type. Write Π G , Π H for the respective fundamental groups of G, H. Thus, we obtain a semi-graph of  [cf. Proposition 2.5, (iii), in the anabelioids of pro-Primes PSC-type H case where Σ = Primes]. Let z G ∈ VCN(G), z H ∈ VCN(H), Π z G ⊆ Π G a VCN-subgroup of Π G associated to z G ∈ VCN(G), Π z H ⊆ Π H a VCN- subgroup of Π H associated to z H ∈ VCN(H), ∼ α  : Π G −→ Π H COMBINATORIAL ANABELIAN TOPICS IV 73 an isomorphism of groups, and γ  ∈ Π H  an element of the [profinite]  Let us fix an injection Π H → Π  such fundamental group Π H  of H. H that the induced outer injection is the outer injection of Proposition 2.5, (iii), and regard subgroups of Π H as subgroups of Π H  by means of this fixed injection. Write Π z H ⊆ Π H  for the closure of Π z H in Π H  . [Thus,  = Π z H ⊆ Π H  is a VCN-subgroup of Π H  associated to z H ∈ VCN( H) VCN(H) — cf. Proposition 2.5, (v).] Then the following hold: (i) It holds that Π z H = Π z H ∩ Π H . (ii) Suppose that α  (Π z G ) ⊆ γ  · Π z H · γ  −1 . Then there exists an element δ ∈ Π H such that α  (Π z G ) ⊆ δ · Π z H · δ −1 . Proof. First, let us observe that it follows immediately from Defini- tion 2.3, (ii), together with the well-known structure of topological fun- damental groups of topological surfaces, that Π z G and Π z H are finitely generated. Thus, it follows immediately from Theorem 2.15 that, to complete the verification of Corollary 2.17, it suffices to verify that the following assertion holds: If Π z H  = Π H , then Π z H is of infinite index in Π H  . To this end, let us observe that since Π z H  = Π H , it follows from Lemma 2.16 [in the case where “G” is a semi-graph of temperoids of HSD-type] that either z H is an edge, or Node(H)  = ∅. On the other hand, in either of these two cases, it follows immediately from Lemma 2.16 [in the case where “G” is a semi-graph of anabelioids of PSC-type], together with Proposition 2.5, (v), that Π z H is of infinite  index in Π H  . This completes the proof of Corollary 2.17. Corollary 2.18 (Properties of VCN-subgroups). Let G be a semi- graph of temperoids of HSD-type. Write Π G for the fundamental group of G. Also, write G  → G for the universal covering of G corresponding to Π G . Then the following hold:  [cf. Definition 2.1, (v)]. Write (i) For i = 1, 2, let v  i ∈ Vert( G) Π v  i ⊆ Π G for the verticial subgroup of Π G associated to v  i [cf. Definition 2.6, (ii)]. Consider the following three [mutually exclusive] conditions [cf. Definition 2.1, (v)]: (1) δ( v 1 , v  2 ) = 0. (2) δ( v 1 , v  2 ) = 1. (3) δ( v 1 , v  2 ) ≥ 2. Then we have equivalences (1) ⇐⇒ (1  ); (2) ⇐⇒ (2  ); (3) ⇐⇒ (3  ) with the following three conditions: 74 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (1  ) Π v  1 = Π v  2 . (2  ) Π v  1 ∩ Π v  2  = {1}, but Π v  1  = Π v  2 . (3  ) Π v  1 ∩ Π v  2 = {1}. (ii) In the situation of (i), suppose that condition (2), hence also v 1 ) ∩ E( v 2 ))  = 1 condition (2  ), holds. Then it holds that (E( [cf. Definition 2.1, (v)], and, moreover, if we write e  ∈ E( v 1 ) ∩ E( v 2 ) for the unique element of E( v 1 ) ∩ E( v 2 ), then Π v  1 ∩ Π v  2 = Π e  ; Π e   = Π v  1 ; Π e   = Π v  2 .  [cf. Definition 2.1, (v)]. Write (iii) For i = 1, 2, let e  i ∈ Edge( G) Π e  i ⊆ Π G for the edge-like subgroup of Π G associated to e  i [cf. Definition 2.6, (ii)]. Then Π e  1 ∩ Π e  2  = {1} if and only if e  1 = e  2 . In particular, Π e  1 ∩Π e  2  = {1} if and only if Π e  1 = Π e  2 [cf. Remark 2.6.1].  e  ∈ Edge( G).  Write Π v  , Π e  ⊆ Π G for the (iv) Let v  ∈ Vert( G), VCN-subgroups of Π G associated to v  , e  , respectively. Then Π e  ∩ Π v   = {1} if and only if e  ∈ E( v ). In particular, Π e  ∩ Π v   = {1} if and only if Π e  ⊆ Π v  [cf. Remark 2.6.1]. (v) Every VCN-subgroup of Π G is commensurably terminal in Π G . Proof. Write G  ∧ → G  for the universal profinite étale covering of the semi-graph of anabelioids of pro-Primes PSC-type G  [cf. Proposition 2.5, (iii), in the case where Σ = Primes] determined by G  → G and Π G  for the [profinite] fundamental group of G  determined by the universal  Thus, one verifies easily that one obtains a nat- covering G  ∧ → G. ural morphism of [pro-]semi-graphs of temperoids [cf. Remark 2.1.1] G  → G  ∧ that induces injections Π G → Π G  [cf. Proposition 2.5, (iii)]  → VCN( G  ∧ ) [cf. [NodNon], Definition 1.1, (iii)] such and VCN( G) that  → VCN( G  ∧ ) is compatible with the re- • the injection VCN( G) spective “δ’s” [cf. Definition 2.1, (v); [NodNon], Definition 1.1, (viii)], and, moreover,  the closure Π z  ⊆ Π  of the image of the • for each z  ∈ VCN( G), G VCN-subgroup Π z  ⊆ Π G of Π G associated to z  via the injection Π G → Π G  coincides with the VCN-subgroup of Π G  [cf. [CbTpI], Definition 2.1, (i)] associated to the image of z  via the injection  → VCN( G  ∧ ) [cf. also Proposition 2.5, (v)]. VCN( G) First, we verify assertion (i). The equivalence (1) ⇔ (1  ) follows im- mediately from the equivalence (1) ⇔ (1  ) of [NodNon], Lemma 1.9, (ii), together with the discussion at the beginning of the present proof. Next, let us observe that, by considering the edge-like subgroup asso- ciated to an element of E( v 1 ) ∩ E( v 2 ), we conclude that condition (2) implies the condition that Π v  1 ∩ Π v  2  = {1}. Thus, the implication (2) COMBINATORIAL ANABELIAN TOPICS IV 75 ⇒ (2  ) follows immediately from the equivalence (1) ⇔ (1  ). The im- plication (2  ) ⇒ (2) follows immediately from Corollary 2.17, (i), and the implication (2  ) ⇒ (2) of [NodNon], Lemma 1.9, (ii), together with the discussion at the beginning of the present proof. The equivalence (3) ⇔ (3  ) follows immediately from the equivalences (1) ⇔ (1  ) and (2) ⇔ (2  ). This completes the proof of assertion (i). Assertion (iii) (respectively, (iv)) follows immediately from [NodNon], Lemma 1.5 (respectively, [NodNon], Lemma 1.7), together with the discussion at the beginning of the present proof. Assertion (v) fol- lows formally from assertions (i), (iii) [cf. also the proof of [CmbGC], Proposition 1.2, (ii)]. Finally, we verify assertion (ii). Suppose that condition (2) [in the statement of assertion (i)], hence also condition (2  ) [in the statement v 2 ))  = 1 of assertion (i)], holds. Then the assertion that (E( v 1 ) ∩ E( follows immediately from the fact that the underlying semi-graph of G  is a tree. The remainder of assertion (ii) follows immediately — in light of assertion (iii) — from Corollary 2.17, (i), and [NodNon], Lemma 1.9, (i) [cf. also Remark 2.6.1], together with the discussion at the beginning of the present proof. This completes the proof of assertion (ii), hence also of Corollary 2.18.  Corollary 2.19 (Graphicity of outer isomorphisms). Let G, H  H  for the semi- be semi-graphs of temperoids of HSD-type. Write G, graphs of anabelioids of pro-Primes PSC-type determined by G, H [cf. Proposition 2.5, (iii), in the case where Σ = Primes], respectively; Π G , Π H for the respective fundamental groups of G, H; Π G  , Π H  for the  H.  Let respective [profinite] fundamental groups of G, ∼ α : Π G −→ Π H ∼ be an outer isomorphism. Write α  : Π G  → Π H  for the outer isomor- phism determined by the outer isomorphism α and the natural outer ∼ ∼  G →  H → isomorphisms Π Π G  , Π Π H  of Proposition 2.5, (iii). Then the following hold: (i) The outer isomorphism α is group-theoretically verticial (respectively, group-theoretically cuspidal; group-theore- tically nodal; graphic) [cf. Definition 2.7, (i), (ii)] if and only if α  is group-theoretically verticial [cf. [CmbGC], Definition 1.4, (iv)] (respectively, group-theoretically cusp- idal [cf. [CmbGC], Definition 1.4, (iv)]; group-theoretically nodal [cf. [NodNon], Definition 1.12]; graphic [cf. [CmbGC], Definition 1.4, (i)]). (ii) The outer isomorphism α is graphic if and only if α is group- theoretically verticial, group-theoretically cuspidal, and group-theoretically nodal. 76 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Proof. Assertion (ii) follows immediately, in light of Corollary 2.18, from a similar argument to the argument applied in the proof of [CmbGC], Proposition 1.5, (ii). Thus, it remains to verify assertion (i). The neces- sity portion of assertion (i) follows immediately from Proposition 2.5, (v). Next, let us observe that inclusions of verticial subgroups of the fundamental group of a semi-graph of temperoids of HSD-type are nec- essarily equalities [cf. Corollary 2.18, (i), (ii)]; a similar statement holds concerning inclusions of edge-like subgroups [cf. Corollary 2.18, (iii)]. Thus, the sufficiency portion of assertion (i) follows immediately — in light of assertion (ii) and [CmbGC], Proposition 1.5, (ii) — from Corollary 2.17, (ii). This completes the proof of Corollary 2.19.  Corollary 2.20 (Discrete combinatorial cuspidalization). Let Σ ⊆ Primes be a subset which is either equal to Primes or of cardinal- ity one, (g, r) a pair of nonnegative numbers such that 2g − 2 + r > 0, n a positive integer, and X a topological surface of type (g, r) [i.e., the complement of r distinct points in an orientable compact topological surface of genus g]. For each positive integer i, write X i for the i-th configuration space of X [i.e., the topological space obtained by forming the complement of the various diagonals in the direct product of i copies of X ]; Π i for the topological fundamental group of X i ; Π Σ i for the pro-Σ  completion of Π i ; Π i for the profinite completion of Π i ; Out FC (Π i ) ⊆ Out F (Π i ) ⊆ Out(Π i ) for the subgroups of the group Out(Π i ) of outomorphisms of Π i defined in the statement of [CmbCsp], Corollary 5.1 [cf. also the discussion entitled “Topological groups” in [CbTpI], §0]; F Σ Σ Out FC (Π Σ i ) ⊆ Out (Π i ) ⊆ Out(Π i ) Σ for the subgroups of the group Out(Π Σ i ) of outomorphisms of Π i con- sisting of FC-admissible, F-admissible [cf. [CmbCsp], Definition 1.1, (ii); the discussion entitled “Topological groups” in [CbTpI], §0] outo- morphisms, respectively. Then the following hold: (i) The group Π n is normally terminal in Π Σ n [cf. Proposition 2.5, (iii)]. In particular, the natural homomorphism Out F (Π n ) −→ Out F (Π Σ n ) is injective. In the following, we shall regard subgroups of Out F (Π n ) as subgroups of Out F (Π Σ n ). F FC  (ii) It holds that Out (Π n ) ∩ Out ( Π n ) = Out FC (Π n ). COMBINATORIAL ANABELIAN TOPICS IV 77 (iii) Consider the commutative diagram  n+1 ) Out F (Π n+1 ) −−−→ Out F ( Π ⏐ ⏐ ⏐ ⏐    n ) Out F (Π n ) −−−→ Out F ( Π — where the horizontal arrows are the injections of (i), and the vertical arrows are the homomorphisms induced by the pro- jection X n+1 → X n obtained by forgetting the (n + 1)-st factor. Suppose that the right-hand vertical arrow of the diagram is injective [cf. Remark 2.20.1 below]. Then the commutative diagram of the above display is cartesian. In particular, the left-hand vertical arrow of the diagram is injective. (iv) The image of the left-hand vertical arrow of the commuta- tive diagram of (iii) [where we do not impose the assumption that the right-hand vertical arrow be injective] is contained in Out FC (Π n ) ⊆ Out F (Π n ). (v) Consider the commutative diagram  n+1 ) Out FC (Π n+1 ) −−−→ Out FC ( Π ⏐ ⏐ ⏐ ⏐    n ) Out FC (Π n ) −−−→ Out FC ( Π — where the horizontal arrows are the injections induced by the injections of (i), and the vertical arrows are the homo- morphisms induced by the projection X n+1 → X n obtained by forgetting the (n + 1)-st factor. This diagram is cartesian, its right-hand vertical arrow is injective, and its left-hand verti- cal arrow is bijective. (vi) Write ⎧ if (g, r) = (0, 3), ⎨ 2 def 3 if (g, r)  = (0, 3) and r  = 0, n FC = ⎩ 4 if r = 0. Suppose that n ≥ n FC . Then it holds that Out FC (Π n ) = Out F (Π n ); the left-hand vertical arrow Out F (Π n+1 ) −→ Out F (Π n ) of the commutative diagram of (iii) is bijective. Proof. Let us first observe that, to verify assertion (i), it suffices to verify that Π n is normally terminal in Π Σ n . Moreover, once one proves the desired normal terminality in the case where n = 1, the desired normal terminality in the case where n ≥ 2 follows immediately by 78 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI induction [cf. the proof of [CmbCsp], Corollary 5.1, (i)]. Thus, we conclude that, to verify assertion (i), it suffices to verify the normal terminality of Π 1 in Π Σ 1 . Next, we claim that the following assertion holds: Claim 2.20.A: Let F be a free nonabelian group. Then F is normally terminal in the pro-Σ completion of F . Indeed, since F is conjugacy l-separable [cf. [Prs], Theorem 3.2] for every l ∈ Σ, Claim 2.20.A follows from a similar argument to the argument applied in the proof of [André], Lemma 3.2.1. This completes the proof of Claim 2.20.A. Next, let us observe that one verifies easily that there exist a semi- graph of temperoids of HSD-type G and an isomorphism of Π 1 with the fundamental group Π G of G. In the following, we shall identify Π G with Π 1 by means of such an isomorphism. If G has a cusp, then it follows from Remark 2.5.1 that Π 1 is a free nonabelian group. Thus, the desired normal terminality follows from Claim 2.20.A. In the remainder of the proof of assertion (i), suppose that G has no cusp. In particular, we may assume without loss of generality, by applying a suitable specialization outer isomorphism [cf. Proposition 2.10], that G has a node. Let γ  ∈ N Π Σ1 (Π 1 ) be an element of the normalizer of Π 1 in Π Σ and Π ⊆ Π v G a 1 verticial subgroup of Π G . Then, by applying Corollary 2.17, (ii) [i.e., in the case where we take the “(G, H, Π z H , Π z G , γ  )” of Corollary 2.17 to be  ) and the “ α ” of Corollary 2.17 to be the automorphism (G, G, Π v , Π v , γ  ], we conclude immediately [cf. also of Π G obtained by conjugation by γ Corollary 2.18, (i), (ii)] that we may assume without loss of generality, by multiplying γ  by a suitable element of Π G , that the element γ  ∈ Π Σ 1 normalizes Π v , hence also the closure Π v of Π v in Π Σ . In particular, 1 it follows from Proposition 2.5, (v); [CmbGC], Proposition 1.2, (ii), that γ  ∈ Π v . On the other hand, since G has a node, it follows from Proposition 2.5, (iv), and Remark 2.6.1 that Π v is a free nonabelian group, and Π v may be identified with the pro-Σ completion of Π v . Thus, it follows from Claim 2.20.A that γ  ∈ Π v ⊆ Π G , as desired. This completes the proof of assertion (i). Assertion (ii) follows immediately from Corollary 2.19, (i). Next, we verify assertion (iii). Let us first observe that since [we have assumed that] the right-hand vertical arrow of the diagram of assertion (iii) is injective, it follows immediately from assertion (i) that all arrows of the diagram of assertion (iii) are injective. Let α ∈ Out F (Π n ) be such that  n ) lies in the image of the right-hand vertical the image of α in Out F ( Π arrow of the diagram of assertion (iii). Then it follows from [CbTpI],  n ) is FC-admissible. Theorem A, (ii), that the image of α in Out F ( Π FC Thus, it follows from assertion (ii) that α ∈ Out (Π n ). In particular, it follows from [NodNon], Corollary 6.6, that there exists a uniquely COMBINATORIAL ANABELIAN TOPICS IV 79 determined element of Out FC (Π n+1 ) whose image in Out F (Π n ) coin- cides with α ∈ Out F (Π n ). Thus, since all arrows of the diagram of assertion (iii) are injective [as verified above], we conclude that the diagram of assertion (iii) is cartesian. This completes the proof of as- sertion (iii). Assertion (iv) follows immediately from [CbTpI], Theorem A, (ii), together with assertion (ii). Assertion (v) follows immediately from a similar argument to the argument applied in the proof of asser- tion (iii), together with the injectivity portion of [NodNon], Theorem B. Assertion (vi) follows immediately from [CbTpII], Theorem A, (ii), together with assertions (i), (ii), (v). This completes the proof of Corol- lary 2.20.  Remark 2.20.1. It follows from [CbTpII], Theorem A, (i), that if either n  = 1 or r  = 0, then the right-hand vertical arrow of the diagram of Corollary 2.20, (iii), is injective. Remark 2.20.2. In the notation of Corollary 2.20, the bijectivity of the left-hand vertical arrow Out FC (Π n+1 ) → Out FC (Π n ) of the diagram of Corollary 2.20, (v), is proven in [NodNon], Corollary 6.6, by apply- ing, in essence, a well-known result concerning topological surfaces due to Dehn-Nielsen-Baer [cf. the proof of [CmbCsp], Corollary 5.1, (ii)]. On the other hand, the equivalences of Corollary 2.19, (i) [cf. also the injection of Corollary 2.20, (i)], together with a similar argument to the argument applied in the proof of the bijectivity portion of [NodNon], Theorem B — i.e., in essence, the argument applied in the proof of [CmbCsp], Corollary 3.3 — allow one to give a purely algebraic alter- native proof of this bijectivity result in the case where n ≥ max{3, n FC } [cf. Corollary 2.20, (vi)]. Corollary 2.21 (Discrete/profinite Dehn multi-twists). In the situation of Example 2.4, (i), write G  X log for the semi-graph of an- abelioids of pro-Primes PSC-type of Proposition 2.5, (iii), in the case where we take “(G, Σ)” to be (G X log , Primes); Π G X log , Π G  log for the X  G for the profi- respective fundamental groups of G X log , G  X log ; Π X log nite completion of Π G X log [so we have a natural outer isomorphism ∼  G → Π G  log — cf. Proposition 2.5, (iii)]; Π X log X Dehn(G X log ) ⊆ Out(Π G X log ) for the subgroup consisting of the Dehn multi-twists of G X log , i.e., of α ∈ Out(Π G X log ) such that the following conditions are satisfied: (a) α is graphic [cf. Definition 2.7, (ii)] and induces the identity automorphism on the underlying semi-graph of G X log . 80 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (b) Let Π v ⊆ Π G X log be a verticial subgroup of Π G X log . Then the outomorphism of Π v induced by restricting α [cf. (a); Corol- lary 2.18, (v); the evident discrete analogue of [CbTpII], Lemma 3.10] is trivial. Then the following hold: (i) The composite of natural outer homomorphisms ∼  G −→ Π G  log Π G X log −→ Π X log X determines an injection Out(Π G X log ) → Out(Π G  log ). X (ii) If one regards subgroups of Out(Π G X log ) as subgroups of Out(Π G  log ) X by means of the injection of (i), then the equality Dehn(G X log ) = Dehn( G  X log ) ∩ Out(Π G X log ) [cf. [CbTpI], Definition 4.4] holds. (iii) The homomorphism of the final display of Example 2.4, (i), de- log (C)| s ) termines, relative to the natural outer isomorphism π 1 (X an ∼ → Π G X log , an isomorphism ∼ log π 1 (S an (C)) −→ Dehn(G X log ) of free Z-modules of rank Node(G X log )  . Moreover, the image of this isomorphism is dense, relative to the profinite topology, in Dehn( G  X log ). Proof. Assertion (i) follows from Corollary 2.20, (i). Next, we verify assertion (ii). The inclusion Dehn(G X log ) ⊆ Dehn( G  X log ) ∩ Out(Π G X log ) follows immediately from the various definitions involved. To verify the reverse inclusion, let α ∈ Dehn( G  X log ) ∩ Out(Π G X log ). Then it follows immediately from Corollary 2.19, (i), together with the definition of Dehn( G  X log ), that the outomorphism α of Π G X log satisfies the condition (a) in the statement of Corollary 2.21. Moreover, it follows immediately from Proposition 2.5, (v), and Corollary 2.20, (i), together with the definition of Dehn( G  X log ), that the outomorphism α of Π G X log satisfies the condition (b) in the statement of Corollary 2.21. This completes the proof of assertion (ii). Finally, we verify assertion (iii). First, let us observe that it fol- lows immediately from the various definitions involved that the ho- momorphism of the final display of Example 2.4, (i), factors through Dehn(G X log ) and has dense image [i.e., relative to the profinite topology] in Dehn( G  X log ) [cf. [CbTpI], Proposition 5.6, (ii)]. Next, let us recall from [CbTpI], Theorem 4.8, (ii), (iv), that if, for e ∈ Node(G X log ) = def Node( G  X log ), we write S e = Node(G X log ) \ {e} and (G X log ) ∧ S e for the semi-graph of anabelioids of pro-Primes PSC-type of Proposition 2.5, COMBINATORIAL ANABELIAN TOPICS IV 81 (iii), in the case where we take “(G, Σ)” to be ((G X log ) S e , Primes) [cf. Definition 2.9] and regard Dehn((G X log ) ∧ S e ) as a closed subgroup of Dehn( G  X log ) via the specialization outer isomorphism of [CbTpI], Definition 2.10 [cf. also Remark 2.9.1, Proposition 2.10 of the present paper], then we have an equality  Dehn((G X log ) ∧ S e ) Dehn( G  X log ) = e∈Node(G X log )  — where each direct summand is [noncanonically] isomorphic to Z. Here, we note that these specialization outer isomorphisms are compat- ible [cf. [CbTpI], Proposition 5.6, (ii), (iii), (iv)] with the corresponding homomorphisms of the final display of Example 2.4, (i). Thus, in light of the density assertion that has already been verified, one verifies im- mediately that, to complete the verification of assertion (iii), it suffices to verify that the image of Dehn(G X log ) via the projection to any di- rect summand of the direct sum decomposition of the above display is contained in some submodule of the direct summand that is isomor- phic to Z. To this end, let us recall from [CbTpI], Theorem 4.8, (iv), that such an image via a projection to a direct summand may be com- puted by considering the homomorphism of the first display of [CbTpI], Lemma 4.6, (ii), i.e., which determines an isomorphism between the di-  e rect summand under consideration and any profinite nodal subgroup Π associated to the node e corresponding to the direct summand. On the other hand, it follows immediately — in light of the definition of this isomorphism — from Proposition 2.5, (v); Corollary 2.17, (i), that the image of Dehn(G X log ) under consideration is contained in a suitable discrete nodal subgroup Π e ( ∼ = Z) associated to e [cf. Remark 2.6.1]. This completes the proof of assertion (iii).  Definition 2.22. Suppose that Σ = Primes. Let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; def def def k = C; S log = Spec(k) log the log scheme obtained by equipping S = Spec(k) with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over S log . For each [possibly empty] subset E ⊆ {1, . . . , n}, write X E log for the E  -th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in [CbTpI], §0], where we think of the factors as being labeled by the elements of E ⊆ {1, . . . , n} [cf. the discussion at the beginning of [CbTpII], §3, in the case where (Σ, k) = (Primes, C)]. For each nonnegative integer n and each [possibly empty] log for the morphism of fs log subset E ⊆ {1, . . . , n}, write (X E log ) an → S an 82 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI analytic spaces determined by the morphism X E log → S log ; (X E log ) an (C), log (C) for the respective topological spaces “X log ” defined in [KN], S an (1.2), in the case where we take the “X” of [KN], (1.2), to be (X E log ) an , log log [cf. the notation established in Example 2.4, (i)]. Let s ∈ S an (C). S an Write def X E = (X E log ) an (C)| s log for the fiber of the natural morphism (X E log ) an (C) → S an (C) at s; def Π disc = π 1 (X E ) E for the discrete topological fundamental group of X E ; def def def = Π disc X n = X {1,...,n} ; X = X 1 ; Π disc n {1,...,n} . Thus, for sets E  ⊆ E ⊆ {1, . . . , n}, we have a projection p an E/E  : X E → X E  obtained by forgetting the factors that belong to E \ E  . For sets E  ⊆ E ⊆ {1, . . . , n} and nonnegative integers m ≤ n, write disc disc disc p Π E/E  : Π E  Π E  for some fixed surjection [that belongs to the collection of surjections that constitutes the outer surjection] induced by p an E/E  ; def disc Π disc Π disc E/E  = Ker(p E/E  ) ⊆ Π E def an p an n/m = p {1,...,n}/{1,...,m} : X n −→ X m ; disc def disc Π disc  Π disc p Π m ; n/m = p {1,...,n}/{1,...,m} : Π n def disc disc Π disc n/m = Π {1,...,n}/{1,...,m} ⊆ Π n .  disc ” for the profinite completion of “Π disc ”. Finally, we shall write “ Π (−) (−) Thus, we have a natural outer isomorphism ∼  disc −→ Π Π E E — where Π E is as in the discussion at the beginning of [CbTpII], §3. def def log ; Π n = Π {1,...,n} . In the following, we shall also write X n log = X {1,...,n} Definition 2.23. In the notation of Definition 2.22, let i ∈ E ⊆ {1, . . . , n}; x ∈ X n (C) a C-valued geometric point of the underlying scheme X n of X n log . (i) We shall write G disc for the semi-graph of temperoids of HSD-type associated to X log [cf. Example 2.4, (ii)]; disc G i∈E,x COMBINATORIAL ANABELIAN TOPICS IV 83 for the semi-graph of temperoids of HSD-type associated to the geometric fiber [cf. Example 2.4, (ii); Remark 2.4.1] of the log log log log projection p log E/(E\{i}) : X E → X E\{i} over x E\{i} → X E\{i} [cf. [CbTpII], Definition 3.1, (i)]; Π G disc , Π G i∈E,x disc disc for the respective fundamental groups of G disc , G i∈E,x [cf. Propo- sition 2.5, (i)];  G disc Π i∈E,x for the profinite completion of Π G i∈E,x disc . Thus, it follows from the discussion of Remark 2.5.2 that we have a natural graphic [cf. [CmbGC], Definition 1.4, (i)] outer isomorphism ∼  G disc −→ Π Π G i∈E,x i∈E,x — where G i∈E,x is the semi-graph of anabelioids of pro-Primes PSC-type of [CbTpII], Definition 3.1, (iii) — and hence a nat- ural isomorphism of semi-graphs of anabelioids ∼ G  disc −→ G i∈E,x i∈E,x disc — where we write G  i∈E,x for the semi-graph of anabelioids of pro-Primes PSC-type of Proposition 2.5, (iii), in the case disc , Primes). Moreover, it where we take “(G, Σ)” to be (G i∈E,x follows immediately from the discussion of Example 2.4 that we have a natural Π disc E -orbit [i.e., relative to composition with automorphisms induced by conjugation by elements of Π disc E ] of isomorphisms ∼ disc (Π disc disc . E ⊇) Π E/(E\{i}) −→ Π G i∈E,x One verifies immediately from the various definitions involved that the diagram ∼  disc  disc Π E/(E\{i}) −−−→ Π G i∈E,x ⏐ ⏐ ⏐ ⏐   ∼ Π E/(E\{i}) −−−→ Π G i∈E,x  disc - — where the upper horizontal arrow is an element of the Π E disc orbit of isomorphisms induced by the Π E -orbit of isomor- phisms of the above discussion; the lower horizontal arrow is an element of the Π E -orbit of isomorphisms of [CbTpII], Def- inition 3.1, (iii); the left-hand vertical arrow is the isomor- phism obtained by forming the restriction of an isomorphism ∼  disc → Π Π E that belongs to the outer isomorphism of the fi- E nal display of Definition 2.22; the right-hand vertical arrow is an isomorphism that belongs to the outer isomorphism of the 84 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI above discussion — commutes up to composition with auto- morphisms induced by conjugation by elements of Π E . disc (ii) We shall say that a vertex v ∈ Vert(G i∈E,x ) is a(n) [E-]tripod of X n if v is of type (0, 3) [cf. Definition 2.6, (iii)]. Thus, one disc ) is a(n) [E-]tripod if and verifies easily that v ∈ Vert(G i∈E,x only if the corresponding vertex of G i∈E,x via the graphic outer ∼  G disc → isomorphism Π Π G i∈E,x of (i) is a(n) [E-]tripod of X n log i∈E,x [cf. [CbTpII], Definition 3.1, (v)]. We shall refer to a verticial subgroup of Π G i∈E,x associated to a(n) [E-]tripod of X n as a(n) disc disc [E-]tripod of Π n . (iii) Let P be a property of [E-]tripods of Π n [cf. [CbTpII], Defi- nition 3.3, (i)] or X n log [e.g., the property of being strict — cf. [CbTpII], Definition 3.3, (iii); the property of arising from an edge — cf. [CbTpII], Definition 3.7, (i); the property of being central — cf. [CbTpII], Definition 3.7, (ii)]. Then we shall say that a(n) [E-]tripod of Π disc or X n [cf. (ii)] satisfies P if the n corresponding [E-]tripod of Π n or X n log satisfies P. (iv) Let T ⊆ Π disc be an E-tripod of Π disc [cf. (ii)]. Then one may n E define the subgroups Out C (T ), Out C (T ) cusp , Out C (T ) Δ , Out C (T ) Δ+ ⊆ Out(T ) of Out(T ) in an entirely analogous fashion to the definition of the closed subgroups “Out C (T )”, “Out C (T ) cusp ”, “Out C (T ) Δ ”, “Out C (T ) Δ+ ” of “Out(T )” given in [CbTpII], Definition 3.4, (i). We leave the routine details to the reader. Theorem 2.24 (Outomorphisms preserving tripods). In the no- tation of Definition 2.22, let E ⊆ {1, . . . , n} be a subset and T ⊆ Π disc E an E-tripod of Π disc [cf. Definition 2.23, (ii)]. Let us write n F disc Out F (Π disc n )[T ] ⊆ Out (Π n ) for the subgroup of Out F (Π disc n ) [cf. the notational conventions intro- duced in the statement of Corollary 2.20] consisting of α ∈ Out F (Π disc n ) disc such that the outomorphism of Π E determined by α preserves the disc Π disc E -conjugacy class of T ⊆ Π E ; def F FC FC disc disc disc Out FC (Π disc n )[T ] = Out (Π n )[T ] ∩ Out (Π n ) ⊆ Out (Π n ) [cf. the notational conventions introduced in the statement of Corol- def def def C disc ) = Out FC (Π disc ); lary 2.20]; Π = Π 1 ; Π disc = Π disc 1 ; Out (Π def Out C (Π) = Out FC (Π). Then the following hold: (i) Write T  for the profinite completion of T . Then the natural homomorphism Out(T ) −→ Out( T  ) COMBINATORIAL ANABELIAN TOPICS IV 85 is injective. If, moreover, one regards subgroups of Out(T ) as subgroups of Out( T  ) via this injection, then it holds that Out C (T ) = Out C ( T  ) ∩ Out(T ), Out C (T ) cusp = Out C ( T  ) cusp ∩ Out(T ), Out C (T ) Δ = Out C ( T  ) Δ ∩ Out(T ), Out C (T ) Δ+ = Out C ( T  ) Δ+ ∩ Out(T ) [cf. Definition 2.23, (iv); [CbTpII], Definition 3.4, (i)]. (ii) It holds that Out C (T ) cusp = Out C (T ) Δ = Out C (T ) Δ+ ∼ = Z/2Z, Out C (T ) ∼ = Z/2Z × S 3 — where we write S 3 for the symmetric group on 3 letters. (iii) The commensurator and centralizer of T ∈ Π disc satisfy the E equality C Π disc (T ) = T × Z Π disc (T ). E E Thus, by applying the evident discrete analogue of [CbTpII], Lemma 3.10, to outomorphisms of Π disc determined by ele- E F disc ments of Out (Π n )[T ], one obtains a natural homomorphism T T : Out F (Π disc n )[T ] −→ Out(T ). (iv) Suppose that n ≥ 3, and that T is central [cf. Definition 2.23, (iii)]. Then it holds that F disc Out F (Π disc n ) = Out (Π n )[T ]. Moreover, the homomorphism F disc T T : Out F (Π disc n ) = Out (Π n )[T ] −→ Out(T ) of (iii) determines a surjection C Δ+ ∼ ( = Z/2Z). Out FC (Π disc n )  Out (T ) We shall refer to this homomorphism as the tripod homo- morphism associated to Π disc n . (v) The profinite completion T  determines an E-tripod of Π n , which, by abuse of notation, we denote by T  . Now suppose that T is E-strict [cf. Definition 2.23, (iii)]. Then it holds that F F disc  Out F (Π disc n )[T ] = Out (Π n )[ T ] ∩ Out (Π n ) [cf. [CbTpII], Theorem 3.16]. 86 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (vi) Suppose that the semi-graph of anabelioids of pro-Primes PSC- type G associated to X log [cf. [CbTpII], Definition 3.1, (ii)] is totally degenerate [cf. [CbTpI], Definition 2.3, (iv)]. Re- call that G may be naturally identified with the semi-graph of anabelioids of pro-Primes PSC-type determined by G disc [cf. Proposition 2.5, (iii); the discussion of Definition 2.23, (i)]. Then one has an equality Aut(G disc ) − = Aut(G) ∩ Out C (Π disc ) − (⊆ Out C (Π)) — where the superscript “ − ’s” denote the closure in the profi- nite topology — of subgroups of Out C (Π) [cf. Corollary 2.20, (i)]. Proof. First, we verify assertion (i). The injectivity portion of asser- tion (i) follows from Corollary 2.20, (i). The first equality follows from Corollary 2.20, (ii). Thus, the second and third equalities follow imme- diately from the various definitions involved; the fourth equality follows from Corollary 2.20, (v). This completes the proof of assertion (i). Next, we verify assertion (ii). The inclusions Out C (T ) Δ+ ⊆ Out C (T ) Δ ⊆ Out C (T ) cusp follow from assertion (i), together with [CbTpII], Lemma 3.5. The inclusion Out C (T ) cusp ⊆ Out C (T ) Δ+ and the assertion that Out C (T ) cusp ∼ = Z/2Z follow immediately from [CmbCsp], Corollary 5.3, (i), together with a classical result of Nielsen [cf. [CmbCsp], Remark 5.3.1]. This completes the proof of the first line of the display of as- sertion (ii). Now since Out C (T ) Δ = Out C (T ) cusp , by considering the action of Out C (T ) on the set of the T -conjugacy classes of cuspidal inertia subgroups of T , we obtain an exact sequence 1 −→ Out C (T ) Δ −→ Out C (T ) −→ S 3 −→ 1. By considering outomorphisms of T arising from automorphisms of analytic spaces, one obtains a section of this sequence; moreover, it follows from the definition of Out C (T ) Δ that this section determines ∼ an isomorphism Out C (T ) Δ × S 3 → Out C (T ). This completes the proof of assertion (ii). Next, we verify assertion (iii). Recall that every finite index subgroup of T is normally terminal in its profinite completion [cf. Corollary 2.20, (i)] and center-free [cf. Remark 2.6.1]. Thus, assertion (iii) follows immediately from [CbTpII], Theorem 3.16, (i). This completes the proof of assertion (iii). Next, we verify assertion (iv). First, let us observe that it fol- lows immediately from the definition of the notion of a central tri- pod [cf. Definition 2.23, (iii); [CbTpII], Definition 3.7, (ii)] that we may assume without loss of generality that n = 3. To verify the equality of the first display of assertion (iv), we mimick the argu- ment in the profinite case given in the proof of [CmbCsp], Corollary  ∈ Aut(Π disc 1.10, (i): Let α ∈ Out F (Π disc n ), α n ) a lifting of α. Write COMBINATORIAL ANABELIAN TOPICS IV 87 α  2 ∈ Aut(Π disc  . Now observe that 2 ) for the automorphism induced by α F disc since α ∈ Out (Π n ), it follows immediately from Corollary 2.20, (iv),  2 preserves that α  2 determines an element of Out FC (Π disc 2 ), hence that α the Π disc -conjugacy class of inertia groups associated to the diagonal 2 [cf. Definition 2.22; the discussion of cusp of any of the fibers of p an 2/1 [CmbCsp], Remark 1.1.5]. Thus, by replacing α  by the composite of α  with a suitable inner automorphism, we may assume without loss of generality that α  2 preserves the inertia group associated to some F disc diagonal cusp of a fiber of p an 2/1 . Now the fact that α ∈ Out (Π n )[T ] follows immediately from Corollary 2.17, (ii); [CbTpII], Theorem 1.9, (ii) [cf. the application of [CmbCsp], Proposition 1.3, (iv), in the proof of [CmbCsp], Corollary 1.10, (i)]. The assertion that the restriction to F disc Out FC (Π disc n ) of the homomorphism Out (Π n ) → Out(T ) of assertion (iii) factors through Out C (T ) Δ+ ⊆ Out(T ) follows immediately from from assertions (i) and (ii), together with [CbTpII], Theorem 3.16, (v). The assertion that the resulting homomorphism is surjective fol- lows immediately from the fact that the [unique] nontrivial element of Out C (T ) Δ+ is the outomorphism induced by complex conjugation [cf. [CmbCsp], Remark 5.3.1], together with the [easily verified] fact that the pointed stable curve over C corresponding to the given stable log curve X log may be assumed, without loss of generality — i.e., by apply- ing a suitable specialization isomorphism [cf. the discussion preceding [CmbCsp], Definition 2.1, as well as [CbTpI], Remark 5.6.1] and ob- serving that such specialization isomorphisms are compatible with the various discrete fundamental groups involved [cf. Remarks 2.9.1 and 2.10.1] — to be defined over R. This completes the proof of assertion (iv). Next, we verify assertion (v). It follows immediately from the clas- sification of E-strict tripods given in [CbTpII], Lemma 3.8, (ii), that we may assume without loss of generality that E  = n ≤ 3. When n = 3, assertion (v) follows formally from assertion (iv). When n = 1, assertion (v) follows immediately from Corollary 2.17, (ii). Thus, it remains to consider the case where n = 2, i.e., where the tripod T arises from an edge. In this case, assertion (v) follows from a similar argument to the argument applied in the proof of assertion (iv). That is to say, let α ∈ Out F (Π disc  ∈ Aut(Π disc 2 ), α 2 ) a lifting of α. Write disc α  1 ∈ Aut(Π 1 ) for the automorphism induced by α  ; β  1 ∈ Aut(Π 1 ),  . Then we must β  ∈ Aut(Π 2 ) for the automorphisms determined by α F disc verify that α ∈ Out (Π 2 )[T ] under the assumption that β  deter- mines an element β ∈ Out F (Π 2 )[ T  ]. Now observe that it follows im- mediately from the computation of the centralizer given in [CbTpII], Lemma 3.11, (vii), that β  1 preserves the Π 1 -conjugacy class of edge-like subgroups of Π 1 determined by the edge that gives rise to the tripod T . Thus, we conclude from Corollary 2.17, (ii), that, by replacing α  88 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI by the composite of α  with a suitable inner automorphism, we may corre- assume that α  1 preserves a specific edge-like subgroup of Π disc 1 sponding to the edge that gives rise to the tripod T . Note that this assumption implies, in light of the commensurably terminality of edge- like subgroups [cf. [CmbGC], Proposition 1.2, (ii)], that β  preserves the Π 2/1 -conjugacy class of the tripod T  . In particular, we conclude, as in the proof of assertion (iv), i.e., by applying Corollary 2.17, (ii), that α ∈ Out F (Π disc 2 )[T ], as desired. This completes the proof of assertion (v). Finally, we verify assertion (vi). First, let us observe that it follows immediately from Corollary 2.20, (v), that both sides of the equality FC C − in question are ⊆ Out FC (Π disc 3 ) ⊆ Out (Π 3 ) (⊆ Out (Π)). Also, we observe that, by considering the case where X log is defined over R [cf. the proof of assertion (iv)], it follows immediately that both sides of the equality in question surject, via the tripod homomorphism of assertion (iv), onto the finite group of order two that appears as the image of this tripod homomorphism [cf. also the fact that the topological group Out( T  ) is profinite, hence, in particular, Hausdorff]. In particular, to complete the proof of assertion (v), it suffices to verify that the evident inclusion Aut(G disc ) − ∩Out FC (Π 3 ) geo ⊆ Aut(G) ∩ Out C (Π disc ) − ∩ Out FC (Π 3 ) geo — where we write Out FC (Π 3 ) geo ⊆ Out FC (Π 3 ) for the kernel of the tripod homomorphism on Out FC (Π 3 ) [cf. [CbTpII], Definition 3.19] — of subgroups of Out C (Π) is, in fact, an equality. On the other hand, since Dehn(G) is a normal open subgroup of both Aut(G disc ) − ∩ Out FC (Π 3 ) geo and Aut(G) ∩ Out C (Π disc ) − ∩ Out FC (Π 3 ) geo [cf. Corol- lary 2.21, (iii); [CbTpI], Theorem 4.8, (i); the commutative diagram of [CbTpII], Corollary 3.27, (ii)], and Aut(G disc ) − ∩ Out FC (Π 3 ) geo clearly surjects onto the finite group of automorphisms of the underlying semi- graph of G disc , the desired equality follows immediately from [CbTpII], Corollary 3.27, (ii). This completes the proof of assertion (vi).  Remark 2.24.1. It is not clear to the authors at the time of writing whether or not one can remove the strictness assumption imposed in Theorem 2.24, (v). Indeed, from the point of view of induction on n, the essential difficulty in removing this assumption may already be seen in the case of a non-E-strict tripod when E  = n = 2. From another point of view, this difficulty may be thought of as arising from the lack of an analogue for discrete topological fundamental groups of n-th configuration spaces, when n ≥ 2, of Corollary 2.17. COMBINATORIAL ANABELIAN TOPICS IV 89 Remark 2.24.2. (i) In the notation of Theorem 2.24, let us observe that it follows from Corollary 2.19, (i), that we have an equality Aut(G disc ) = Aut(G) ∩ Out C (Π disc ) (⊆ Out C (Π)) of subgroups of Out C (Π) [cf. Corollary 2.20, (i)]. On the other hand, it is by no means clear whether or not the evident inclu- sion Aut(G disc ) − ⊆ Aut(G) ∩ Out C (Π disc ) − (⊆ Out C (Π)) (∗) — where the superscript “ − ’s” denote the closure in the profi- nite topology — is an equality in general. On the other hand, when X log is totally degenerate, this equality is the content of Theorem 2.24, (vi). (ii) We continue to use the notation of (i). Write M Q for the moduli stack of hyperbolic curves of type (g, r) over Q and C Q → M Q for the tautological hyperbolic curve over M Q . Thus, def for appropriate choices of basepoints, if we write Π C = π 1 (C Q ), def Π M = π 1 (M Q ) for the respective étale fundamental groups, then we obtain an exact sequence of profinite groups 1 −→ Δ C/M −→ Π C −→ Π M −→ 1 — where Δ C/M is defined so as to render the sequence exact — as well as a natural outer representation ρ M : Π M −→ Out C (Π) — where, by choosing appropriate basepoints, we identify Π with Δ C/M — and a natural outer surjection Π M  G Q onto the absolute Galois group G Q of Q [cf. the discussion of [CbTpII], Remark 3.19.1]. Write G R ⊆ G Q for the decomposi- tion group [which is well-defined up to G Q -conjugation] of the unique archimedean prime of Q. In the spirit of [Bgg1], [Bgg2], [Bgg3], let us write def Γ = Out C (Π disc ) (⊆ Out C (Π)); def Γ̌ = ρ M (Π M × G Q G R ) [cf. Corollary 2.20, (i)]. Thus, for appropriate choices of base- points, Γ̌ is equal to the closure of Γ in Out C (Π). If σ is a sim- plex of the complex of profinite curves L(Π) studied in [Bgg1], [Bgg2], [Bgg3], that arises from Π disc , then the stabilizer in Γ of σ is denoted Γ σ , while the stabilizer in Γ̌ of the image of σ in the profinite curve complex corresponding to Γ̌ is denoted Γ̌ σ . Then [Bgg3], Theorem 4.2 [cf. also [Bgg1], Proposition 6.5], asserts that 90 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI The natural inclusion Γ − σ ⊆ Γ̌ σ is, in fact, an equality. Translated into the language of the present paper, this asser- tion corresponds precisely to the assertion that the inclusion (∗) considered in (i) is, in fact, an equality. In particular, The- orem 2.24, (vi), corresponds, essentially, to a special case [i.e., the totally degenerate case] of [Bgg3], Theorem 4.2. At a more concrete level, when Node(G)  = 1, and σ arises from a single simple closed curve that corresponds to the unique node e of G, this assertion corresponds precisely to the assertion that the profinite stabilizer in Γ̌ of the Π-conjugacy class of nodal subgroups of Π determined by e coin- cides with the closure in Γ̌ of the discrete stabilizer in Γ of the Π disc -conjugacy class of nodal subgroups of Π disc determined by e — cf. Theorem 3.3, Remark 3.3.1, Corollary 3.4 in §3 below. As discussed in (i), this sort of assertion is highly nontriv- ial. That is to say, this sort of coincidence between a profinite stabilizer and the closure of a corresponding discrete stabilizer is, in fact, false in general, as the example given in (iv) be- low demonstrates. In particular, this sort of coincidence is by no means a consequence of superficial “general nonsense”-type considerations, but rather, when true [cf., e.g., the case treated in Theorem 2.24, (vi)], a consequence of deep properties of the specific groups and specific spaces [on which these groups act] under consideration. (iii) In closing, we observe that many of the results derived in [Bgg3] as a consequence of the assertion discussed in (ii) were, in fact, already obtained in earlier papers by the authors. Indeed, the faithfulness asserted in [Bgg3], Theorem 7.7 — i.e., the injectivity of the restriction of ρ M to a section G F → Π M arising from a hyperbolic curve of type (g, r) defined over a number field F — is a special case of [NodNon], Theorem C. On the other hand, in [CbTpI], Theorem D, a computation is given of the centralizer in Out C (Π) of an open subgroup of Γ̌. Thus, the computation of centers given in [Bgg3], Corollary 6.2, amounts to a special case of [CbTpI], Theorem D. Finally, [Bgg3], Corollary 7.6 — which may be regarded as the assertion that the inverse image via ρ M of the centralizer of Γ̌ in Out C (Π) maps trivially to G Q — amounts to a concatenation of • the computation of the centralizer given in [CbTpI], The- orem D, with • the fact, stated in [NodNon], Corollary 6.4, that ρ −1 M (Γ̌) maps trivially to G Q . COMBINATORIAL ANABELIAN TOPICS IV 91 (iv) Let n ≥ 3 be an integer. Consider the natural conjuga- tion action of the special linear group SL n (Z) with coefficients ∈ Z on the module M n (Z) of n by n matrices with coefficients ∈ Z. Write A ∈ M n (Z) for the diagonal matrix whose entries are given by the integers 1, . . . , n. Then one verifies immedi- ately that the stabilizer SL n (Z) A of A, relative to the conjugacy action of SL n (Z), is equal to the subgroup of diagonal matrices of SL n (Z), hence isomorphic to the finite group given by a product of n − 1 copies of the finite group of order two {±1}. On the other hand, if one considers  with coefficients the action of the special linear group SL n ( Z)   ∈ Z on the module M n ( Z) of n by n matrices with coefficients  then one verifies immediately that the stabilizer ∈ Z,  A SL n ( Z)  is equal to of A, relative to the conjugacy action of SL n ( Z),  hence isomorphic the subgroup of diagonal matrices of SL n ( Z),  × , a group of uncountable to a product of n − 1 copies of Z cardinality. That is to say,  A is much larger The profinite stabilizer SL n ( Z) than the profinite completion of the discrete stabi- lizer SL n (Z) A . Here, we recall that since, as is well-known, the congruence subgroup problem has been resolved in the affirmative, in the  may be identified case of n ≥ 3, the topological group SL n ( Z) with the profinite completion of the group SL n (Z). A simi- lar example may be given in the case of the symplectic group Sp 2n (Z). Corollary 2.25 (Characterization of the archimedean local Ga- lois groups in the global Galois image associated to a hyper- bolic curve). Let F be a number field [i.e., a finite extension of the field of rational numbers]; p an archimedean prime of F ; F p an algebraic closure of the p-adic completion F p of F [so F p is isomor- phic to C]; F ⊆ F p the algebraic closure of F in F p ; X F log a smooth def def log curve over F . Write G p = Gal(F p /F p ) ⊆ G F = Gal(F /F ); def def def = X F log × F F p ; X F log = X F log × F F p ; X F log = X F log × F F ; X F log p p π 1 (X F log ) 92 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI for the log fundamental group of X F log ; π 1 disc (X F log ) p for the [discrete] topological fundamental group of the analytic space associated to the interior of the log scheme X F log ; p π 1 disc (X F log ) ∧ p for the profinite completion of π 1 disc (X F log ); p ρ X log : G F −→ Out(π 1 (X F log )) F for the natural outer Galois action associated to X F log ; ρ disc : G p −→ Out(π 1 disc (X F log )) X log ,p p F . Thus, we have for the natural outer Galois action associated to X F log p a natural outer isomorphism ∼ π 1 disc (X F log ) ∧ −→ π 1 (X F log ), p which determines a natural injection Out(π 1 disc (X F log )) → Out(π 1 (X F log )) p [cf. Corollary 2.20, (i)]. Then the following hold: (i) We have a natural commutative diagram ρ disc log X ,p ρ log F G p −−− −→ Out(π 1 disc (X F log )) p ⏐ ⏐ ⏐ ⏐   X F G F −−− → Out(π 1 (X F log )) — where the vertical arrows are the natural inclusions, and all arrows are injective. (ii) The diagram of (i) is cartesian, i.e., if we regard the various groups involved as subgroups of Out(π 1 (X F log )), then we have an equality G p = G F ∩ Out(π 1 disc (X F log )). p Proof. Assertion (i) follows immediately from the injectivity of the lower horizontal arrow ρ X log [cf. [NodNon], Theorem C], together with F the various definitions involved. Finally, we verify assertion (ii). Write (X F ) log 3 for the 3-rd log con- log figuration space of X F . Then it follows immediately from [NodNon], Theorem B, that the group Out FC (π 1 ((X F ) log 3 )) of FC-admissible outo- log morphisms of the log fundamental group π 1 ((X F ) log 3 ) of (X F ) 3 may be COMBINATORIAL ANABELIAN TOPICS IV 93 regarded as a closed subgroup of Out(π 1 (X F log )). Moreover, it follows immediately from the various definitions involved that the respective images Im(ρ X log ), Im(ρ disc ) of the natural outer Galois actions ρ X log , X log ,p F F F associated to X F log , X F log are contained in this closed subgroup ρ disc log p ,p X F log Out FC (π 1 ((X F ) log 3 )) ⊆ Out(π 1 (X F )). Thus, to verify assertion (ii), one verifies immediately from Corollary 2.20, (v), that it suffices to verify the equality ) = Im(ρ X log ) ∩ Out(π 1 disc ((X F p ) log Im(ρ disc 3 )) X log ,p F F def log log disc — where we write (X F p ) log 3 = (X F ) 3 × F F p and π 1 ((X F p ) 3 ) for the [discrete] topological fundamental group of the analytic space associated to the interior of the log scheme (X F p ) log 3 . On the other hand, since the “ρ X log ” that occurs in the case where we take “X F log ” to be the F smooth log curve associated to P 1 F \ {0, 1, ∞} is injective [cf. assertion (i)], this equality follows immediately — by considering the images of the subgroups ) ⊆ Im(ρ X log ) ∩ Out(π 1 disc ((X F p ) log Im(ρ disc 3 )) X log ,p F F of Out(π 1 disc ((X F p ) log 3 )) via the [manifestly compatible!] tripod homo- morphisms associated to π 1 disc ((X F p ) log 3 ) [cf. Theorem 2.24, (iv)] and log π 1 ((X F ) 3 ) [cf. [CbTpII], Theorem 3.16, (i), (v)] — from [André], The- orem 3.3.1. This completes the proof of assertion (ii), hence also of Corollary 2.25.  Remark 2.25.1. Corollary 2.25 is a generalization of [André], Theo- rem 3.3.2 [cf. also the footnote of [André] following [André], Theorem 3.3.2]. Although the proof given here of Corollary 2.25 is by no means the first proof of this result [cf. the discussion of this footnote of [André] following [André], Theorem 3.3.2; [NodNon], Corollary 6.4], it is of in- terest to note that this result may also be derived in the context of the theory of the present paper, i.e., via an argument that parallels the proof given in [CbTpIII] of [CbTpIII], Theorem B, in the p-adic case [for which no alternative proofs are known!]. 94 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI 3. Canonical liftings of cycles In the present §3, we discuss certain canonical liftings of cycles [cf. Theorems 3.10, 3.14 below]. These canonical liftings are constructed in a fashion illustrated in Figure 1. This approach to constructing such canonical liftings was motivated [cf. Remark 3.10.1 below] by the ar- guments of [Bgg2], where these canonical liftings were applied, in the context of the congruence subgroup problem for hyperelliptic modular groups, to derive certain injectivity results [cf. [Bgg2], §2], which may be regarded as special cases of [NodNon], Theorem B. Unfortunately, however, the authors of the present paper were unable to follow in detail these arguments of [Bgg2], which appear to be based to a substantial extent on geometric intuition concerning the geometry of topological surfaces. Although, in the development of the present series of pa- pers on combinatorial anabelian geometry, the authors were motivated by similar geometric intuition, the proofs of the results given in the present series of papers proceed by means of purely combinatorial and algebraic arguments concerning combinatorial [e.g., graphs] and group- theoretic [e.g., profinite fundamental groups] data that arises from a pointed stable curve. From the point of view of arithmetic geome- try, the geometric intuition which underlies the topological arguments given in [Bgg2] involving objects such as topological Dehn twists is of an essentially archimedean nature, hence, in particular, is fundamentally incompatible, at least from the point of view of establishing a rigorous mathematical formulation, with the highly nonarchimedean properties of profinite fundamental groups, as studied in the present series of papers — cf. the discussion of [SemiAn], Remark 1.5.1. It was this state of affairs that motivated the authors to give, in the present §3, a formulation of the constructions of [Bgg2], §2, in terms of the purely combinatorial and algebraic techniques developed in the present series of papers. In the present §3, let (g, r) be a pair of nonnegative integers such that 2g − 2 + r > 0; n a positive integer; Σ a set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one; k def an algebraically closed field of characteristic ∈ Σ; S log = Spec(k) log the def log scheme obtained by equipping S = Spec(k) with the log structure determined by the fs chart N → k that maps 1 → 0; X log = X 1 log a stable log curve of type (g, r) over S log . For each [possibly empty] subset E ⊆ {1, . . . , n}, write X E log for the E  -th log configuration space of the stable log curve X log [cf. the discussion entitled “Curves” in [CbTpI], §0], where we think of the factors as being labeled by the elements of E ⊆ {1, . . . , n}; Π E COMBINATORIAL ANABELIAN TOPICS IV 95 for the maximal pro-Σ quotient of the kernel of the natural surjection π 1 (X E log )  π 1 (S log ); log log Π  p log E/E  : X E → X E  , p E/E  : Π E  Π E , def def def log log Π E/E  = Ker(p Π E/E  ) ⊆ Π E , X n = X {1,...,n} , Π n = Π {1,...,n} , def log log log p log n/m = p {1,...,n}/{1,...,m} : X n −→ X m , def Π p Π n/m = p {1,...,n}/{1,...,m} : Π n  Π m , def Π n/m = Π {1,...,n}/{1,...,m} ⊆ Π n , G, G, Π G , G i∈E,x , Π G i∈E,x for the objects defined in the discussion at the beginning of [CbTpII], §3; [CbTpII], Definition 3.1. In addition, we suppose that we have been given a pair of nonnegative integers ( Y g, Y r) such that 2 Y g − 2 + Y r > 0 and a stable log curve Y log = Y 1 log of type ( Y g, Y r) over S log . We shall use similar notation log log Y Π p E/E  : Y Π E  Y Π E  , Y E log , Y Π E , Y p log E/E  : Y E → Y E  , Y def def def log Y log Π E/E  = Ker( Y p Π = Y {1,...,n} , Y Π n = Y Π {1,...,n} , E/E  ) ⊆ Π E , Y n Y log def Y log p n/m = p {1,...,n}/{1,...,m} : Y n log −→ Y m log , def Y Π Y Y p n/m = Y p Π {1,...,n}/{1,...,m} : Π n  Π m , def Π n/m = Y Π {1,...,n}/{1,...,m} ⊆ Y Π n , Y G, Y G, Π Y G , Y G i∈E,y , Π Y G i∈E,y Y for objects associated to the stable log curve Y log = Y 1 log to the nota- tion introduced above for X log [cf. the discussion at the beginning of [CbTpII], §3; [CbTpII], Definition 3.1]. Lemma 3.1 (Graphicity in the case of a single node). In the notation of the discussion at the beginning of the present §3, suppose that Node(G)  = Node( Y G)  = 1. Write e ∈ Node(G) (respectively, Y e ∈ Node( Y G)) for the unique node of G (respectively, Y G). Let Π e ⊆ Π 1 (respectively, ∼ ∼ Π Y e ⊆ Y Π 1 ) be a nodal subgroup of Π 1 → Π G (respectively, Y Π 1 → Π Y G ) associated to e ∈ Node(G) (respectively, Y e ∈ Node( Y G)); e 2 ∈ X 2 (k) (respectively, Y e 2 ∈ Y 2 (k)) a k-valued point of the underlying scheme X 2 (respectively, Y 2 ) of the log scheme X 2 log (respectively, Y 2 log ) that Y log lies, relative to p log 2/1 (respectively, p 2/1 ), over the k-valued point of X (respectively, Y ) determined by the node e ∈ Node(G) (respectively, Y e ∈ Node( Y G)). Thus, we obtain an outer isomorphism ∼ Π 2/1 −→ Π G 2∈{1,2},e 2 ∼ (respectively, Y Π 2/1 → Π Y G 2∈{1,2},Y e ) 2 96 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. [CbTpII], Definition 3.1, (iii)] that may be characterized, up to composition with elements of Aut |grph| (G 2∈{1,2},e 2 ) ⊆ Out(Π G 2∈{1,2},e 2 ) (respectively, Aut |grph| ( Y G 2∈{1,2}, Y e 2 ) ⊆ Out(Π Y G 2∈{1,2},Y e )) [cf. [CbTpI], 2 Definition 2.6, (i); [CbTpII], Remark 4.1.2], as the group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)] outer isomorphism such that the semi-graph of anabelioids structure on G 2∈{1,2},e 2 (respectively, Y G 2∈{1,2}, Y e 2 ) is the semi-graph of anabelioids structure determined [cf. [NodNon], Theorem A] by the resulting composite outer representa- tion ∼ Π e → Π 1 → Out(Π 2/1 ) → Out(Π G 2∈{1,2},e 2 ) ∼ (respectively, Π Y e → Y Π 1 → Out( Y Π 2/1 ) → Out(Π Y G 2∈{1,2},Y e )) 2 — where the second arrow is the outer action determined by the exact sequence 1 → Π 2/1 → Π 2 → Π 1 → 1 (respectively, 1 → Y Π 2/1 → Y Π 2 → Y Π 1 → 1) — in a fashion compatible with the restriction Π 2/1  Π {2} Y Π (respectively, Y Π 2/1  Y Π {2} ) of p Π {1,2}/{2} (respectively, p {1,2}/{2} ) to Π 2/1 ⊆ Π 2 (respectively, Y Π 2/1 ⊆ Y Π 2 ) and the given outer isomor- ∼ ∼ ∼ ∼ phisms Π {2} → Π 1 → Π G (respectively, Y Π {2} → Y Π 1 → Y Π G ). Let (respectively, Y v ∈ Vert( Y G 2∈{1,2}, Y e 2 )) v ∈ Vert(G 2∈{1,2},e 2 ) be the {1, 2}-tripod [cf. [CbTpII], Definition 3.1, (v)] that arises from e ∈ Node(G) (respectively, Y e ∈ Node( Y G)) [cf. [CbTpII], Defini- ∼ tion 3.7, (i)]; Π v ⊆ Π G 2∈{1,2},e 2 ← Π 2/1 (respectively, Π Y v ⊆ Π Y G 2∈{1,2},Y e 2 ∼ ← Y Π 2/1 ) a {1, 2}-tripod in Π 2 (respectively, Y Π 2 ) associated to the tri- pod v (respectively, Y v) [cf. [CbTpII], Definition 3.3, (i)]; ∼ α : Π G −→ Π Y G an outer isomorphism of profinite groups. Suppose that the following conditions are satisfied: (a) The outer isomorphism α is group-theoretically nodal [cf. [NodNon], Definition 1.12], i.e., determines a bijection of the set of Π G -conjugates of Π e ⊆ Π G and the set of Π Y G -conjugates of Π Y e ⊆ Π Y G . (b) The outer isomorphism α is 2-cuspidalizable [cf. [CbTpII], Definition 3.20], i.e., the outer isomorphism ∼ α ∼ ∼ Π 1 −→ Π G −→ Π Y G ←− Y Π 1 arises from a [uniquely determined, up to permutation of the 2 factors — cf. [NodNon], Theorem B] PFC-admissible [cf. ∼ [CbTpI], Definition 1.4, (iii)] outer isomorphism Π 2 → Y Π 2 . [In particular, the outer isomorphism α is group-theoretically cuspidal.] Then the following hold: COMBINATORIAL ANABELIAN TOPICS IV 97 ∼ (i) There exists a PFC-admissible isomorphism α  2 : Π 2 → Y Π 2 that lifts α such that the composite ∼ ∼ ∼ Π G 2∈{1,2},e 2 ←− Π 2/1 −→ Y Π 2/1 −→ Π Y G 2∈{1,2},Y e 2 — where the second arrow is the restriction of α  2 — is graphic [cf. [CmbGC], Definition 1.4, (i)]. ∼ (ii) The outer isomorphism α 2 : Π 2 → Y Π 2 determined by the iso- morphism α  2 of (i) induces a bijection between the set of Π 2 - conjugates of Π v ⊆ Π 2 and the set of Y Π 2 -conjugates of Π Y v ⊆ Y Π 2 . Moreover, if we think of Π v , Π Y v as the respective [pro-Σ] fundamental groups of G 2∈{1,2},e 2 | v , Y G 2∈{1,2}, Y e 2 | Y v [cf. [CbTpI], Definition 2.1, (iii); [CbTpI], Remark 2.1.1], then the induced ∼ outer isomorphism Π v → Π Y v [cf. [CbTpII], Theorem 3.16, (i)] is group-theoretically cuspidal. (iii) The outer isomorphism α is graphic. Proof. In light of conditions (a) and (b), assertion (i) follows immedi- ately from [NodNon], Theorem A [cf. also our assumption that Node(G)  = Node( Y G)  = 1, which implies that the outer representation Π e → Out(Π G 2∈{1,2},e 2 ) (respectively, Π Y e → Out(Π Y G 2∈{1,2},Y e )) is nodally non- 2 degenerate!]. Next, let us observe that the Π G 2∈{1,2},e 2 - (respectively, Π Y G 2∈{1,2},Y e -) conjugacy class of Π v ⊆ Π G 2∈{1,2},e 2 (respectively, Π Y v ⊆ 2 Π Y G 2∈{1,2},Y e ) may be characterized as the unique Π G 2∈{1,2},e 2 - (respectively, 2 Π Y G 2∈{1,2},Y e -) conjugacy class of verticial subgroups that fails to map 2 injectively via the surjection Π 2/1  Π {2} (respectively, Y Π 2/1  Y Π {2} ). Now assertion (ii) follows immediately from assertion (i). Assertion (iii) follows immediately — in light of [CmbCsp], Proposition 1.2, (iii) — from assertions (i), (ii), together with the various definitions involved. This completes the proof of Lemma 3.1.  Before proceeding, we pause to observe that Lemma 3.1 may be applied to obtain an alternative proof of a slightly weaker version of Theorem 3.3 below, as follows. Proposition 3.2 (Graphicity of group-theoretically nodal 2-cus- pidalizable outer isomorphisms). In the notation of the discussion at the beginning of the present §3, let ∼ α : Π G −→ Π Y G be an outer isomorphism of profinite groups. Suppose that the following conditions are satisfied: (a) The outer isomorphism α is group-theoretically nodal [cf. [NodNon], Definition 1.12]. 98 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (b) The outer isomorphism α is 2-cuspidalizable [cf. [CbTpII], Definition 3.20], i.e., the outer isomorphism α ∼ ∼ ∼ Π 1 −→ Π G −→ Π Y G ←− Y Π 1 arises from a [uniquely determined, up to permutation of the 2 factors — cf. [NodNon], Theorem B] PFC-admissible [cf. ∼ [CbTpI], Definition 1.4, (iii)] outer isomorphism Π 2 → Y Π 2 . [In particular, the outer isomorphism α is group-theoretically cuspidal — cf. [CmbGC], Definition 1.4, (iv).] Then the outer isomorphism α is graphic [cf. [CmbGC], Definition 1.4, (i)]. Proof. Let us first observe that it follows from condition (a), together with [CmbGC], Proposition 1.2, (i), that α determines a bijection ∼ Node(G) → Node( Y G), so Node(G)  = Node( Y G)  . We verify Propo- sition 3.2 by induction on Node(G)  = Node( Y G)  . If Node(G) = Node( Y G) = ∅, then Proposition 3.2 is immediate. Thus, we may assume without loss of generality that Node(G), Node( Y G)  = ∅. Let e ∈ Node(G). Write Y e ∈ Node( Y G) for the node of Y G that corresponds, via α, to e. Write G {e} (respectively, Y G { Y e} ) for the generization of G (respectively, Y G) with respect to {e} ⊆ Node(G) (respectively, { Y e} ⊆ Node( Y G)) [cf. [CbTpI], Definition 2.8]; β for the composite outer isomorphism Φ G {e} ∼ α ∼ Π G {e} −→ Π G −→ Π Y G Φ −1 Y G { Y e} ∼ −→ Π Y G { Y e} [cf. [CbTpI], Definition 2.10]; v 0 ∈ Vert(G {e} ) (respectively, Y v 0 ∈ Vert( Y G { Y e} )) for the [uniquely determined] vertex of the generiza- tion G {e} (respectively, Y G { Y e} ) that does not arise from a vertex of Vert(G) (respectively, Vert( Y G)). Let Π v 0 ⊆ Π G {e} (respectively, Π Y v 0 ⊆ Π Y G { Y e} ) be a verticial subgroup associated to v 0 ∈ Vert(G {e} ) (respectively, Y v 0 ∈ Vert( Y G { Y e} )); Π e ⊆ Π v 0 (respectively, Π Y e ⊆ Π Y v 0 ) a subgroup that maps to a nodal subgroup associated to e in Π G (respectively, to Y e in Π Y G ). Thus, it follows immediately from [NodNon], Lemma 1.9, (i), (ii) [cf. also [NodNon], Lemma 1.5; con- dition (2) of [CbTpI], Proposition 2.9, (i)], that Π v 0 (respectively, Π Y v 0 ) may be characterized as the unique verticial subgroup of Π G {e} (respectively, Π Y G { Y e} ) that contains Π e (respectively, Π Y e ). Next, let us observe that, by applying the induction hypothesis to β, we conclude that β is graphic. Thus, it follows immediately — in light of [CmbGC], Proposition 1.5, (ii) — from the definition of the gener- izations under consideration [cf. condition (3) of [CbTpI], Proposition 2.9, (i)] that, to complete the verification of Proposition 3.2, it suffices to verify that the following assertion holds: COMBINATORIAL ANABELIAN TOPICS IV 99 Claim 3.2.A: Let H ⊆ Π v 0 ⊆ Π G {e} be a closed sub- group of Π v 0 whose image in Π G is a verticial subgroup. Then the image of H via the composite β ∼ Π G {e} −→ Π Y G { Y e} Φ Y G { Y e} ∼ −→ Π Y G is a verticial subgroup. To verify Claim 3.2.A, let us observe that since β is graphic, it fol- lows immediately from the above characterization of Π v 0 , Π Y v 0 that β maps Π v 0 bijectively onto a Π Y G { Y e} -conjugate of Π Y v 0 . Thus, it follows immediately from condition (b), together with the evident iso- morphism [i.e., as opposed to outomorphism — cf. [CbTpII], Remark 4.14.1] version of [CbTpII], Lemma 4.8, (i), (ii), that, in the notation ∼ of [CbTpII], Definition 4.3, the outer isomorphism Π 2 → Y Π 2 of con- ∼ dition (b) induces compatible outer isomorphisms (Π v 0 ) 2 → (Π Y v 0 ) 2 , ∼ Π v 0 → Π Y v 0 . In particular, by applying Lemma 3.1, (iii), to these outer isomorphisms, one concludes that Claim 3.2.A holds, as desired. This completes the proof of Proposition 3.2.  Theorem 3.3 (Graphicity of profinite outer isomorphisms). Let Σ 0 be a nonempty set of prime numbers; H, J semi-graphs of anabe- lioids of pro-Σ 0 PSC-type; Π H , Π J the [pro-Σ 0 ] fundamental groups of H, J , respectively; ∼ α : Π H −→ Π J an outer isomorphism of profinite groups. Then the following condi- tions are equivalent: (i) The outer isomorphism α is graphic [cf. [CmbGC], Definition 1.4, (i)]. (ii) The outer isomorphism α is group-theoretically verticial and group-theoretically cuspidal [cf. [CmbGC], Definition 1.4, (iv)]. (iii) The outer isomorphism α is group-theoretically nodal [cf. [NodNon], Definition 1.12] and group-theoretically cuspi- dal. Proof. The implication (i) ⇒ (ii) (respectively, (ii) ⇒ (iii)) follows from the various definitions involved (respectively, [NodNon], Lemma 1.9, (i)). Thus, it suffices to verify the implication (iii) ⇒ (i). Suppose that condition (iii) holds. Then, to verify the graphicity of α, it follows from [CmbGC], Theorem 1.6, (ii), that it suffices to verify that α is graphically filtration-preserving [cf. [CmbGC], Definition 1.4, (iii)]. In particular, by replacing Π H , Π J by suitable open subgroups of Π H , Π J , 100 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI it suffices to verify that α determines isomorphisms ∼ ∼ Π ab-edge −→ Π ab-edge , Π ab-vert −→ Π ab-vert H J H J — where we write “Π ab-edge ”, “Π ab-vert ” for the closed subgroups of (−) (−) ab the abelianization “Π (−) ” of “Π (−) ” topologically generated by the im- ages of the edge-like, verticial subgroups of “Π (−) ”. Here, we may as- sume without loss of generality that H and J are sturdy, hence admit compactifications [cf. [CmbGC], Remarks 1.1.5, 1.1.6]. Now the asser- ” follows immediately from condition (iii). On tion concerning “Π ab-edge (−) the other hand, the assertion concerning “Π ab-vert ” follows immediately (−) from the duality discussed in [CmbGC], Proposition 1.3, applied to the compactifications of H, J , together with condition (iii). This completes the proof of Theorem 3.3.  Remark 3.3.1. Here, we observe that results such as [Bgg3], Corol- lary 6.1; [Bgg3], Corollary 6.4, (ii); [Bgg3], Theorem 6.6, amount, when translated into the language of the present paper, to a special case of the result obtained by concatenating the equivalence (i) ⇔ (iii) of The- orem 3.3, with the computation of the normalizer given in [CbTpI], Theorem 5.14, (iii) [i.e., in essence, [CmbGC], Corollary 2.7, (iii), (iv)]. Moreover, the proof given above of this equivalence (i) ⇔ (iii) of Theo- rem 3.3 is, essentially, a restatement of various results from the theory of [CmbGC]. That is to say, although the statements of these results that occur in the present series of papers and in [Bgg3] are formulated and arranged in a somewhat different way, the essential mathematical content that underlies these results is, in fact, entirely identical; more- over, this state of affairs is by no means a coincidence. Indeed, this mathematical content is given in [CmbGC] as [CmbGC], Proposition 1.3; [CmbGC], Proposition 2.6. In [Bgg3], this mathematical content is given as [Bgg3], Lemma 5.11 [and the surrounding discussion], which, in fact, was related to the author of [Bgg3] by the senior author of the present paper in the context of an explanation of the theory of [CmbGC]. Corollary 3.4 (Graphicity of discrete outer isomorphisms). Let H, J be semi-graphs of temperoids of HSD-type [cf. Definition 2.3, (iii)]; Π H , Π J the fundamental groups of H, J , respectively [cf. Propo- sition 2.5, (i)]; ∼ α : Π H −→ Π J an outer isomorphism. Then the following conditions are equivalent: (i) The outer isomorphism α is graphic [cf. Definition 2.7, (ii)]. COMBINATORIAL ANABELIAN TOPICS IV 101 (ii) The outer isomorphism α is group-theoretically verticial and group-theoretically cuspidal [cf. Definition 2.7, (i)]. (iii) The outer isomorphism α is group-theoretically nodal and group-theoretically cuspidal [cf. Definition 2.7, (i)]. Proof. This follows immediately from Theorem 3.3, together with Corol- lary 2.19, (i).  ∼ Definition 3.5. Let ( Y G, S ⊆ Node( Y G), φ : Y G S → G) be a degener- ation structure on G [cf. [CbTpII], Definition 3.23, (i)] and e ∈ S. (i) We shall say that a closed subgroup J ⊆ Π 1 of Π 1 is a cycle- ∼ subgroup of Π 1 [with respect to ( Y G, S ⊆ Node( Y G), φ : Y G S → G), associated to e ∈ S] if J is contained in the Π 1 -conjugacy class of closed subgroups of Π 1 obtained by forming the image of a nodal subgroup of Π Y G associated to e via the composite of outer isomorphisms Φ −1 Y G S ∼ ∼ ∼ Π Y G −→ Π Y G S −→ Π G −→ Π 1 — where the first arrow is the inverse of the specialization outer isomorphism Φ Y G S [cf. [CbTpI], Definition 2.10], the second ∼ arrow is the graphic outer isomorphism Π Y G S → Π G induced by φ, and the third arrow is the natural outer isomorphism ∼ Π G → Π 1 of [CbTpII], Definition 3.1, (ii) [cf. the left-hand portion of Figure 1]. (ii) Let n be a positive integer. Then we shall say that a cycle- subgroup of Π 1 is n-cuspidalizable if it is a cycle-subgroup of Π 1 with respect to some n-cuspidalizable degeneration structure on G [cf. [CbTpII], Definition 3.23, (v)]. Remark 3.5.1. Let J ⊆ Π 1 be a cycle-subgroup of Π 1 with respect ∼ to a degeneration structure ( Y G, S ⊆ Node( Y G), φ : Y G S → G), associ- ated to a node e ∈ S. Then it follows immediately from [CmbGC], Proposition 1.2, (i), that the node e of Y G is uniquely determined by the subgroup J ⊆ Π 1 and the degeneration structure ( Y G, S ⊆ ∼ Node( Y G), φ : Y G S → G). Definition 3.6. Let J ⊆ Π 1 be a 2-cuspidalizable cycle-subgroup of Π 1 [cf. Definition 3.5, (i), (ii)]. (i) It follows immediately from the various definitions involved that we have data as follows: 102 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (a) a 2-cuspidalizable degeneration structure ( Y G, S ⊆ Node( Y G), ∼ φ : Y G S → G) on G [cf. [CbTpII], Definition 3.23, (i), (v)], ∼ (b) an isomorphism Y Π 1 → Π 1 that is compatible with the composite of the display of Definition 3.5, (i) [cf. also [CbTpII], Definition 3.1, (ii)], in the case where we take ∼ the “( Y G, S ⊆ Node( Y G), φ : Y G S → G)” of Definition 3.5 to be the degeneration structure of (a), ∼ (c) a PFC-admissible isomorphism Y Π 2 → Π 2 that lifts the isomorphism of (b), and (d) a nodal subgroup Π e ⊆ Y Π 1 of Y Π 1 associated to a [uniquely determined — cf. Remark 3.5.1] node e of Y G such that the image of the nodal subgroup Π e ⊆ Y Π 1 of (d) ∼ via the isomorphism Y Π 1 → Π 1 of (b) coincides with J ⊆ Π 1 . We shall say that a closed subgroup T ⊆ Π 2/1 of Π 2/1 is a tripodal subgroup associated to J if T coincides — relative to some choice of data (a), (b), (c), (d) as above [but cf. also Re- ∼ mark 3.6.1!] — with the image, via the lifting Y Π 2 → Π 2 of (c), of some {1, 2}-tripod in Y Π 2/1 ⊆ Y Π 2 [cf. [CbTpII], Def- inition 3.3, (i)] arising from e [cf. [CbTpII], Definition 3.7, (i)], and, moreover, the centralizer Z Π 2 (T ) maps bijectively, via p Π 2/1 : Π 2  Π 1 , onto J ⊆ Π 1 [cf. [CbTpII], Lemma 3.11, (iv), (vii)]. (ii) Let T ⊆ Π 2/1 be a tripodal subgroup associated to J [cf. (i)]. Then we shall refer to a closed subgroup of T that arises from a nodal (respectively, cuspidal) subgroup contained in the {1, 2}-tripod in Y Π 2/1 ⊆ Y Π 2 of (i) as a lifting cycle-subgroup (respectively, distinguished cuspidal subgroup) of T [cf. the right- hand portion of Figure 1]. Remark 3.6.1. Note that, in the situation of Definition 3.6, (i), it fol- lows immediately from Lemma 3.1, (ii) [i.e., by considering the gener- ization of Y G with respect to Node( Y G) \ {e} — cf. [CbTpI], Definition 2.8], together with the computation of the centralizer given in [CbTpII], Lemma 3.11, (vii), and the commensurable terminality of J ⊆ Π 1 [cf. [CmbGC], Proposition 1.2, (ii)], that the Π 2/1 -conjugacy class of a tripodal subgroup T is completely determined by the cycle-subgroup J ⊆ Π 1 . Remark 3.6.2. (i) Suppose that we are in the situation of Definition 3.5, (i). Re- call the module Λ G , i.e., the cyclotome associated to G, defined in [CbTpI], Definition 3.8, (i). Thus, as an abstract module, COMBINATORIAL ANABELIAN TOPICS IV 103  Σ of Z. Recall, fur- Λ G is isomorphic to the pro-Σ completion Z thermore, from [CbTpI], Corollary 3.9, (v), (vi), that one may construct a natural, functorial {±}-orbit of isomorphisms ∼ Π e −→ Λ Y G ∼ — where Π e ⊆ Y Π 1 → Π Y G [cf. [CbTpII], Definition 3.1, (ii)] denotes a nodal subgroup associated to e. Thus, by apply- ∼ ing the natural, functorial [outer] isomorphisms Λ Y G → Λ Y G S ∼ [cf. [CbTpI], Corollary 3.9, (i)] and Φ −1 : Π Y G → Π Y G S [cf. Y G S [CbTpI], Definition 2.10], together with the [outer] isomor- ∼ ∼ phisms Λ Y G S → Λ G and Π Y G S → Π G induced by φ, we obtain a natural {±}-orbit of isomorphisms ∼ J −→ Λ G associated to the cycle-subgroup J ⊆ Π 1 . Note that this {±}- orbit of isomorphisms is functorial with respect to automor- phisms α of Π 1 such that α(J) = J, and, moreover, the outer automorphism of Π G obtained by forming the conjugate of α ∼ by the natural outer isomorphism Π 1 → Π G is graphic [cf. the equivalence (i) ⇔ (iii) of Theorem 3.3]. In this context, it is natural to refer to either of the two isomorphisms in this {±}-orbit as an orientation on the cycle-subgroup J. (ii) Now suppose that we are in the situation of Definition 3.6, (i), (ii). Then let us observe that the natural outer surjec- ∼ tion Y Π 2/1  Y Π {2} → Y Π 1 determined by Y p Π {1,2}/{2} induces a natural isomorphism ∼ Λ Y G 2∈{1,2},e 2 −→ Λ Y G [cf. [CbTpI], Corollary 3.9, (ii)], where we write e 2 ∈ Y 2 (k) for a k-valued point of Y 2 that lies, relative to Y p log 2/1 , over the k-valued point of Y determined by the node e. Write v for the vertex of Y G 2∈{1,2},e 2 that gives rise to the tripodal subgroup T ⊆ Π 2/1 . Thus, we have a natural isomorphism ∼ Λ v −→ Λ Y G 2∈{1,2},e 2 [cf. [CbTpI], Corollary 3.9, (ii)]. Now suppose that e ∗ is a node of Y G 2∈{1,2},e 2 that abuts to v and, moreover, gives rise to a lifting cycle-subgroup J ∗ ⊆ T of the tripodal subgroup T . Thus, one verifies immediately that the natural outer sur- ∼ jection Π 2/1  Π {2} → Π 1 determined by p Π {1,2}/{2} induces a ∼ natural isomorphism J ∗ → J [cf. [CbTpII], Lemma 3.6, (iv)]. Let Π e ∗ ⊆ Π Y G 2∈{1,2},e 2 be a nodal subgroup associated to e ∗ . Then the [unique!] branch of e ∗ that abuts to v determines a 104 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI natural isomorphism ∼ Π e ∗ −→ Λ v [cf. [CbTpI], Corollary 3.9, (v)]. Thus, by composing the iso- morphisms of the last three displays with the isomorphism ∼ ∼ Λ Y G → Λ Y G S → Λ G discussed in (i) and the inverse of the ∼ tautological isomorphism Π e ∗ → J ∗ , we obtain a natural iso- morphism ∼ J ∗ −→ Λ G associated to the lifting cycle-subgroup J ∗ ⊆ T . Note that this natural isomorphism is functorial with respect to FC- admissible automorphisms α 2 of Π 2 such that α 2 (J ∗ ) = J ∗ , α 2 (T ) = T , and, moreover, the outer automorphism of Π G ob- tained by forming the conjugate, by the natural outer isomor- ∼ phism Π 1 → Π G , of the outer automorphism of Π 1 determined by α 2 is graphic [cf. the equivalence (i) ⇔ (iii) of Theorem 3.3; [CbTpII], Lemma 3.11, (vii)]. Finally, one verifies immediately from the construction of the isomorphisms of [CbTpI], Corol- ∼ lary 3.9, (v), that if one composes this isomorphism J ∗ → Λ G ∼ with the inverse of the natural isomorphism J ∗ → J discussed ∼ above, then the resulting isomorphism J → Λ G is an orienta- tion on the cycle-subgroup J, in the sense of the discussion of (i), and, moreover, that, if we define an orientation on the tripodal subgroup T to be a choice of a T -conjugacy class of lifting cycle-subgroups of T , then the resulting assignment     orientations on T −→ orientations on J is a bijection [between sets of cardinality 2]. Lemma 3.7 (Induced outomorphisms of tripods). In the situa- tion of Lemma 3.1, suppose that X log = Y log . Write c ∈ Cusp(G 2∈{1,2},e 2 ) for the cusp arising from the diagonal divisor in X × k X. Let Π c ⊆ Π G 2∈{1,2},e 2 be a cuspidal subgroup of Π G 2∈{1,2},e 2 associated to c. Write def α v = T Π v (α 2 ) ∈ Out(Π v ) [cf. Lemma 3.1, (ii); [CbTpII], Theorem 3.16, (i)] for the result of applying the tripod homomorphism T Π v to α 2 . [Thus, it follows immediately from Lemma 3.1, (ii), that α v ∈ Out C (Π v ).] Suppose, moreover, that the following condition is satisfied: ∼ (c) The cuspidal subgroup Π c ⊆ Π G 2∈{1,2},e 2 ← Π 2/1 is contained in Π v . Then the following hold: COMBINATORIAL ANABELIAN TOPICS IV 105 (i) Since Π v may be regarded as the “Π 1 ” that occurs in the case where we take “X log ” to be the smooth log curve associated to P 1 k \ {0, 1, ∞} [cf. [CbTpII], Remark 3.3.1], there exists a uniquely determined outomorphism ι ∈ Out(Π v ) of Π v that arises from an automorphism of P 1 k \{0, 1, ∞} over k and induces a nontrivial automorphism of the set N (v). Write def def |α v | = α v ∈ Out(Π v ) (respectively, |α v | = ι◦α v ∈ Out(Π v )) if α v ∈ Out C (Π v ) cusp (respectively, ∈ Out C (Π v ) cusp ) [cf. [CbTpII], Definition 3.4, (i)]. Then it holds that |α v | ∈ Out C (Π v ) cusp . (ii) Let Π tpd ⊆ Π 3 be a central {1, 2, 3}-tripod of Π 3 [cf. [CbTpII], Definitions 3.3, (i); 3.7, (ii)]. Then every geometric [cf. ∼ [CbTpII], Definition 3.4, (ii)] outer isomorphism Π tpd → Π v ∼ satisfies the following condition: Let β ∈ Out(Π 1 ) → Out(Π G ) ∼ be an outomorphism of Π 1 → Π G that is group-theoretically nodal and 3-cuspidalizable, i.e., β ∈ Out(Π 1 ) arises from a(n) [uniquely determined — cf. [NodNon], Theorem B] FC- admissible outomorphism β 3 ∈ Out FC (Π 3 ). Then the image T Π tpd (β 3 ) ∈ Out(Π tpd ) [cf. [CbTpII], Definition 3.19] coincides — relative to the ∼ outer isomorphism Π tpd → Π v under consideration — with |β v | ∈ Out(Π v ) def [cf. (i)], where we write β v = T Π v (β 3 ) ∈ Out(Π v ). In particu- lar, it holds that |β v | ∈ Out C (Π v ) Δ+ [cf. [CbTpII], Definition 3.4, (i)]. Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). Let us first observe that the inclusion |β v | ∈ Out C (Π v ) Δ follows immediately from the coincidence of T Π tpd (β 3 ) with |β v |, relative to some specific geometric outer iso- ∼ morphism Π tpd → Π v , together with the second displayed equality of [CbTpII], Theorem 3.16, (v). The inclusion |β v | ∈ Out C (Π v ) Δ+ then follows from [CbTpII], Lemma 3.5; [CbTpII], Theorem 3.17, (i) [ap- plied in the case where we take the “(Π 2 , T, T  )” of loc. cit. to be (Π 3/1 , T, Π tpd )]. Moreover, it follows immediately from the various def- initions involved that the inclusion |β v | ∈ Out C (Π v ) Δ allows one to conclude that the coincidence of T Π tpd (β 3 ) with |β v |, relative to some ∼ specific geometric outer isomorphism Π tpd → Π v , implies the coinci- dence of T Π tpd (β 3 ) with |β v |, relative to an arbitrary geometric outer ∼ isomorphism Π tpd → Π v . Thus, to complete the verification of assertion (ii), it suffices to verify the coincidence of T Π tpd (β 3 ) with |β v |, relative 106 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∼ to the specific geometric outer isomorphism Π tpd → Π v whose exis- tence is guaranteed by [CbTpII], Theorem 3.18, (ii). In the following ∼ discussion, we fix this specific geometric outer isomorphism Π tpd → Π v . Next, let us observe that if β v = |β v |, i.e., β v ∈ Out C (Π v ) cusp , then it follows immediately from [CbTpII], Theorems 3.16, (v); 3.18, (ii), that T Π tpd (β 3 ) ∈ Out(Π tpd ) coincides with |β v | ∈ Out(Π v ). Thus, to complete the verification of assertion (ii), we may assume without loss of generality that β v  = |β v |, i.e., that β v ∈ Out C (Π v ) cusp . Then let us observe that collections of data consisting of smooth log curves that [by gluing at prescribed cusps] give rise to a stable log curve whose associated semi-graph of anabelioids [of pro-Σ PSC-type] is isomorphic to G may be parametrized by a smooth, connected moduli stack. Thus, one verifies easily that, by considering the étale fundamental groupoid of this moduli stack, together with a suitable scheme-theoretic auto- morphism of order 2 of a collection of data parametrized by this mod- uli stack, one obtains a 3-cuspidalizable automorphism ξ ∈ Aut(G) (→ Out(Π G )) of G such that ξ v [i.e., the “α v ” that occurs in the case where we take “α” to be ξ] coincides with ι. Thus, by applying the portion of assertion (ii) that has already been verified to ξ ◦ β, we con- clude that, to complete the verification of assertion (ii), it suffices to verify that T Π tpd (ξ 3 ) = 1. On the other hand, this follows immediately from the fact that ξ was assumed to arise from a scheme-theoretic au- tomorphism [cf. also [CbTpII], Theorem 3.16, (v)]. This completes the proof of assertion (ii) and hence of Lemma 3.7.  Definition 3.8. Let J ⊆ Π 1 be a 2-cuspidalizable cycle-subgroup [cf. Definition 3.5, (i), (ii)]; let us fix associated data as in Definition 3.6, (i), (a), (b), (c), (d). Relative to this data, suppose that T ⊆ Π 2/1 is a tripodal subgroup associated to J ⊆ Π 1 [cf. Definition 3.6, (i)], and that I ⊆ T is a distinguished cuspidal subgroup of T [cf. Definition 3.6, (ii)]. Note that this data, together with the log scheme structure of Y log , allows one to speak of geometric [cf. [CbTpII], Definition 3.4, (ii)] out- omorphisms of T . Then one verifies easily that there exists a uniquely determined nontrivial geometric outomorphism of T that preserves the T -conjugacy class of I. Thus, since I is commensurably terminal in T [cf. [CmbGC], Proposition 1.2, (ii)], there exists a uniquely determined I-conjugacy class of automorphisms of T that lifts this outomorphism and preserves I ⊆ T . We shall refer to this I-conjugacy class of auto- morphisms of T as the cycle symmetry associated to I. Before proceeding, we pause to observe the following interesting “al- ternative formulation” of the essential content of Lemma 3.7, (ii). COMBINATORIAL ANABELIAN TOPICS IV 107 Lemma 3.9 (Geometricity of conjugates of geometric outer isomorphisms). Suppose that we are in the situation of [CbTpII], Theorem 3.18, (ii), i.e., n ≥ 3, and T (respectively, T  ) is an E- (respectively, E  -) tripod of Π n for some subset E ⊆ {1, . . . , n} (re- ∼ spectively, E  ⊆ {1, . . . , n}). Let φ : T → T  be a geometric [cf. [CbTpII], Definition 3.4, (ii)] outer isomorphism. Then, for every α ∈ Out FC (Π n )[T, T  : {|C|}], the composite of outer isomorphisms T T (α) ∼ φ ∼ T −→ T −→ T  T T  (α) −1 ∼ −→ T  [cf. [CbTpII], Theorem 3.16, (i)] is equal to φ. Proof. Let us first observe that the validity of Lemma 3.9 for some spe- cific geometric outer isomorphism “φ” follows formally from the com- mutative diagram of [CbTpII], Theorem 3.18, (ii). Thus, the validity of Lemma 3.9 for an arbitrary geometric outer isomorphism “φ” follows immediately from the equality of the first display of [CbTpII], Theorem 3.18, (i), i.e., the fact that T T (α) commutes with arbitrary geometric outomorphisms of T . This completes the proof of Lemma 3.9.  Remark 3.9.1. One verifies immediately that a similar argument to the argument applied in the proof of Lemma 3.9 yields evident ana- logues of Lemma 3.9 in the respective situations of [CbTpII], Theorem 3.17, (i), (ii). Theorem 3.10 (Canonical liftings of cycles). In the notation of the discussion at the beginning of the present §3, let I ⊆ Π 2/1 ⊆ Π 2 be a cuspidal inertia group associated to the diagonal cusp of a fiber of p log 2/1 ; Π tpd ⊆ Π 3 a 3-central {1, 2, 3}-tripod of Π 3 [cf. [CbTpII], Definition 3.7, (ii)]; I tpd ⊆ Π tpd a cuspidal subgroup of Π tpd that does ∗ ∗∗ not arise from a cusp of a fiber of p log 3/2 ; J tpd , J tpd ⊆ Π tpd cuspidal ∗ ∗∗ subgroups of Π tpd such that I tpd , J tpd , and J tpd determine three dis- tinct Π tpd -conjugacy classes of closed subgroups of Π tpd . [Note that one verifies immediately from the various definitions involved that such ∗ ∗∗ cuspidal subgroups I tpd , J tpd , and J tpd always exist.] For positive inte- FC gers n ≥ 2, m ≤ n and α ∈ Aut (Π n ) [cf. [CmbCsp], Definition 1.1, (ii)], write α m ∈ Aut FC (Π m ) for the automorphism of Π m determined by α; Aut FC (Π n , I) ⊆ Aut FC (Π n ) for the subgroup consisting of β ∈ Aut FC (Π n ) such that β 2 (I) = I; Aut FC (Π n ) G ⊆ Aut FC (Π n ) 108 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI for the subgroup consisting of β ∈ Aut FC (Π n ) such that the image of β via the composite Aut FC (Π n )  Out FC (Π n ) → Out FC (Π 1 ) → Out(Π G ) — where the second arrow is the natural injection of [NodNon], Theo- rem B, and the third arrow is the homomorphism induced by the natural ∼ outer isomorphism Π 1 → Π G — is graphic [cf. [CmbGC], Definition 1.4, (i)]; def Aut FC (Π n , I) G = Aut FC (Π n , I) ∩ Aut FC (Π n ) G ; Cycle n (Π 1 ) for the set of n-cuspidalizable cycle-subgroups of Π 1 [cf. Defini- tion 3.5, (i), (ii)]; Tpd I (Π 2/1 ) for the set of closed subgroups T ⊆ Π 2/1 such that T is a tripodal sub- group associated to some 2-cuspidalizable cycle-subgroup of Π 1 [cf. Definition 3.6, (i)], and, moreover, I is a distinguished cuspidal subgroup [cf. Definition 3.6, (ii)] of T . Then the following hold: (i) Let n ≥ 2 be a positive integer, α ∈ Aut FC (Π n , I) G , J ∈ Cycle n (Π 1 ), and T ∈ Tpd I (Π 2/1 ). Then it holds that α 1 (J) ∈ Cycle n (Π 1 ), α 2 (T ) ∈ Tpd I (Π 2/1 ). Thus, Aut FC (Π n , I) G acts naturally on Cycle n (Π 1 ), Tpd I (Π 2/1 ). (ii) Let n ≥ 2 be a positive integer. Then there exists a unique Aut FC (Π n , I) G -equivariant [cf. (i)] map C I : Cycle n (Π 1 ) −→ Tpd I (Π 2/1 ) such that, for every J ∈ Cycle n (Π 1 ), C I (J) is a tripodal sub- group associated to J. Moreover, for every α ∈ Aut FC (Π n , I) G ∼ and J ∈ Cycle n (Π 1 ), the isomorphism C I (J) → C I (α 1 (J)) induced by α 2 maps every lifting cycle-subgroup [cf. Def- inition 3.6, (ii)] of C I (J) bijectively onto a lifting cycle- subgroup of C I (α 1 (J)). (iii) Let n ≥ 3 be a positive integer. Then there exists an assign- ment Cycle n (Π 1 ) J → syn I,J — where syn I,J denotes an I-conjugacy class of isomorphisms ∼ Π tpd → C I (J) — such that (a) syn I,J maps I tpd bijectively onto I, ∗ ∗∗ (b) syn I,J maps the subgroups J tpd , J tpd bijectively onto lift- ing cycle-subgroups of C I (J), and COMBINATORIAL ANABELIAN TOPICS IV 109 (c) for α ∈ Aut FC (Π n , I) G , the diagram [of I tpd -, I-conjugacy classes of isomorphisms] Π tpd −−−→ ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J)  1 C I (J) −−−→ C I (α 1 (J)) — where the upper horizontal arrow is the [uniquely de- termined — cf. the commensurable terminality of I tpd in Π tpd discussed in [CmbGC], Proposition 1.2, (ii)] I tpd - conjugacy class of automorphisms of Π tpd that lifts T Π tpd (α) [cf. [CbTpII], Definition 3.19] and preserves I tpd ; the lower horizontal arrow is the I-conjugacy class of isomorphisms induced by α 2 [cf. (ii)] — commutes up to possible com- position with the cycle symmetry of C I (α 1 (J)) associ- ated to I [cf. Definition 3.8]. Finally, the assignment J → syn I,J is uniquely determined, up to possible composition with cy- cle symmetries, by these conditions (a), (b), and (c). (iv) Let n ≥ 3 be a positive integer, α ∈ Aut FC (Π n , I) G , and J ∈ Cycle n (Π 1 ). Suppose that one of the following conditions is satisfied: (a) The FC-admissible outomorphism of Π 3 determined by α 3 is ∈ Out FC (Π 3 ) geo [cf. [CbTpII], Definition 3.19]. (b) Cusp(G)  = ∅. (c) n ≥ 4. Then there exists an automorphism β ∈ Aut FC (Π n , I) G such that the FC-admissible outomorphism of Π 3 determined by β 3 is contained in Out FC (Π 3 ) geo , and, moreover, α 1 (J) = β 1 (J). Finally, the diagram [of I tpd -, I-conjugacy classes of isomor- phisms] Π tpd ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J) =syn I,β (J)  1 1 C I (J) −−−→ C I (α 1 (J)) = C I (β 1 (J)) — where the lower horizontal arrow is the isomorphism induced by β 2 [cf. (ii)] — commutes up to possible composition with the cycle symmetry of C I (α 1 (J)) = C I (β 1 (J)) associated to I. Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). The initial portion of assertion (ii) follows immediately from the discussion of Remark 3.6.1, together 110 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI with the fact that T is uniquely determined among its Π 2/1 -conjugates by the condition I ⊆ T [cf. [CmbGC], Proposition 1.5, (i)]. The final portion of assertion (ii) follows immediately from Lemma 3.1, (ii) [i.e., by considering a suitable generization operation, as in the discussion of Remark 3.6.1]. This completes the proof of assertion (ii). Next, we verify assertion (iii). Let us fix data ∼ ∼ ( Y G, S ⊆ Node( Y G), φ : Y G S → G); Y Π 1 → Π 1 ; Y ∼ Π 2 → Π 2 ; Π e ⊆ Y Π 1 for J ∈ Cycle n (Π 1 ) as in Definition 3.6, (i), (a), (b), (c), (d), and let Y T ⊆ Y Π 2/1 be a {1, 2}-tripod as in the discussion of Definition 3.6, (i). Let Y Π tpd ⊆ Y Π 3 be a 3-central tripod of Y Π 3 . Here, we note that since J ∈ Cycle n (Π 1 ), and n ≥ 3, it follows that the above isomorphism ∼ ∼ Y Π 2 → Π 2 lifts to a PFC-admissible isomorphism Y Π 3 → Π 3 that maps Y Π tpd to a Π 3 -conjugate of Π tpd [cf. [NodNon], Theorem B; [CbTpII], Theorem 3.16, (v); [CbTpII], Remark 4.14.1]. Now one verifies immediately that, to verify the existence portion of assertion (iii), by applying a suitable generization operation as in the discussion of Remark 3.6.1, we may assume without loss of gen- erality that Node( Y G)  = 1 [an assumption that will be invoked when we apply Lemma 3.7 in the argument to follow]. Then, by considering the geometric [hence, in particular, C-admissible] outer isomorphism of [CbTpII], Theorem 3.18, (ii), in the case where we take the “(T, T  )” of [CbTpII], Theorem 3.18, (ii), to be ( Y Π tpd , Y T ), we obtain an outer ∼ isomorphism Π tpd → C I (J). Moreover, by considering the composite of this outer isomorphism with a suitable geometric outomorphism of Π tpd , we may assume without loss of generality that this outer iso- ∼ morphism Π tpd → C I (J) maps the Π tpd -conjugacy class of I tpd to the C I (J)-conjugacy class of I. Thus, since I is commensurably terminal in C I (J) [cf. [CmbGC], Proposition 1.2, (ii)], we obtain a uniquely de- ∼ termined I-conjugacy class of isomorphisms syn I,J : Π tpd → C I (J) that lifts the outer isomorphism just discussed and satisfies condition (a). On the other hand, one verifies immediately from the various definitions involved that syn I,J also satisfies condition (b). Next, we verify that syn I,J satisfies condition (c). To this end, let us observe that it follows immediately from the various definitions involved [cf. also our assumption that Node( Y G)  = 1], that α 1 (J) admits data as in Definition 3.6, (i), (a), (b), (c), (d), such that • the portion of this data that corresponds to the data of Defi- nition 3.6, (i), (a), (d), is of the form ∼ ( Y G, S ⊆ Node( Y G), ψ : Y G S → G); Π e ⊆ Y Π 1 ∼ for some isomorphism ψ : Y G S → G, and, moreover, COMBINATORIAL ANABELIAN TOPICS IV 111 • the composite Y ∼ α 2 ∼ ∼ Π 2 −→ Π 2 −→ Π 2 ←− Y Π 2 — where the first (respectively, third) arrow is the isomor- phism arising from the data [cf. Definition 3.6, (i), (c)] for J (respectively, α 1 (J)) ∈ Cycle n (Π 1 ) under consideration — is the identity automorphism. Thus, to verify the assertion that syn I,J satisfies condition (c), it suffices ∼ to verify that the I-conjugacy class of isomorphisms “syn I,J : Π tpd → C I (J)” constructed above from a fixed choice of data as in Defini- tion 3.6, (i), (a), (b), (c), (d) does not depend on this choice of data. On the other hand, this follows immediately from Lemma 3.7, (ii) [cf. our assumption that Node( Y G)  = 1]. Finally, we consider the final portion of assertion (iii) concerning uniqueness. To this end, we observe that, by considering the case where Y G, as well as each of the branches of the underlying semi-graph of Y G, is defined over a number field F , it follows immediately, by considering automorphisms α ∈ Aut FC (Π n , I) G that arise from scheme theory, that given any element γ ∈ Out(Π tpd ) that arises from an element of the absolute Galois group of F , there exists an α ∈ Aut FC (Π n , I) G such that α(J) = J and T Π tpd (α) = γ. Thus, the uniqueness under consideration follows immediately from the geometricity of elements of Out(Π tpd ) that commute with the image of the absolute Galois group of F , i.e., in other words, from the Grothendieck Conjecture for tripods over number fields [cf. [Tama1], Theorem 0.3; [LocAn], Theorem A]. This completes the proof of assertion (iii). Finally, we verify assertion (iv). If condition (a) is satisfied, then, by taking the “β” of assertion (iv) to be α, we conclude that assertion (iv) follows immediately from assertion (iii), together with the definition of Out FC (Π n ) geo . Next, let us observe that, by applying assertion (iv) in the case where condition (a) is satisfied, we conclude that, to verify assertion (iv) in the case where either (b) or (c) is satisfied, it suffices to verify that the following assertion holds: Claim 3.10.A: Write Out(Π 1 ⊇ J) ⊆ Out(Π 1 ) for the subgroup of Out(Π 1 ) consisting of outomor- phisms of Π 1 that preserve the Π 1 -conjugacy class of J and def Out FC (Π n ) G = Aut FC (Π n ) G /Inn(Π n ) ⊆ Out FC (Π n ). Then every element of the image of the natural injec- tion Out FC (Π n ) G → Out FC (Π 1 ) 112 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI [cf. [NodNon], Theorem B] may be written as a prod- uct of an element of the image of the natural injec- tion Out FC (Π n ) geo → Out FC (Π 1 ) and an element of def Out(Π 1 ⊇ J) G = Out(Π 1 ⊇ J) ∩ Out FC (Π 1 ) G . To verify Claim 3.10.A, write Out FC (Π n , J) G ⊆ Out FC (Π n ) G for the subgroup of Out FC (Π n ) G obtained by forming the inverse image of the closed subgroup Out(Π 1 ⊇ J) ⊆ Out(Π 1 ) via the natural injection Out FC (Π n ) G → Out FC (Π 1 ). Then one verifies immediately, by consid- ering the exact sequence T Πtpd 1 −→ Out FC (Π n ) geo −→ Out FC (Π n ) −→ Out C (Π tpd ) Δ+ −→ 1 [cf. conditions (b), (c); [CbTpII], Definition 3.19; [CbTpII], Corollary 4.15], that, to verify Claim 3.10.A, it suffices to verify that the following assertion holds: Claim 3.10.B: The composite T Πtpd Out FC (Π n , J) G → Out FC (Π n )  Out C (Π tpd ) Δ+ is surjective. ∼ To verify Claim 3.10.B, let ( Y G, S ⊆ Node( Y G), φ : Y G S → G) be an n-cuspidalizable degeneration structure on G with respect to which J is a cycle-subgroup such that Y G is totally degenerate [cf. [CbTpI], Defini- tion 2.3, (iv)]. [One verifies immediately that such a degeneration struc- ture always exists.] Now let us identify Out FC (Π n ) with Out FC ( Y Π n ) via a(n) [uniquely determined, up to permutation of the n factors — cf. [NodNon], Theorem B] PFC-admissible [cf. [CbTpI], Definition 1.4, ∼ (iii)] outer isomorphism Π n → Y Π n that is compatible with the out- omorphism of the display of Definition 3.5, (i) [cf. [CbTpII], Propo- sition 3.24, (i)]. Then it follows immediately from the various defini- tions involved that the closed subgroup Out FC ( Y Π n ) brch ⊆ Out FC ( Y Π n ) [cf. [CbTpII], Definition 4.6, (i)] is contained in the closed subgroup Out FC (Π n , J) G ⊆ Out FC (Π n ). On the other hand, it follows immedi- ately from the proof of [CbTpII], Corollary 4.15, that the composite T Πtpd Out FC ( Y Π n ) brch → Out FC ( Y Π n ) = Out FC (Π n )  Out C (Π tpd ) Δ+ is surjective. This completes the proof of Claim 3.10.B, hence also of assertion (iv) in the case where either (b) or (c) is satisfied.  Remark 3.10.1. (i) The content of Theorem 3.10, (iv), may be regarded, i.e., by considering the various lifting cycle-subgroups involved, as a formulation of the construction of the two sections discussed in [Bgg2], Proposition 2.7 [which plays an essential role in the COMBINATORIAL ANABELIAN TOPICS IV 113 proof of [Bgg2], Theorem 2.4], in terms of the purely combina- torial and algebraic techniques developed in the present series of papers. (ii) In this context, we observe in passing that [one verifies imme- diately that] for arbitrary nonnegative integers g, r such that • 3g − 3 + r > 0, and, moreover, • if g = 0, then r is even, there exists a stable log curve of type (g, r) which admits an automorphism that is linear over the base scheme under con- sideration and fixes a node of the stable log curve, but switches the branches of this node. Thus, by considering the resulting automorphism of the associated semi-graph of anabelioids of pro-Σ PSC-type, one concludes that the diagrams of Theo- rem 3.10, (iii), (iv), fail to commute, in general, if one does not allow for the possibility of composition with a cycle symmetry. This situation contrasts with the situation discussed in [Bgg2], Proposition 2.7, where two independent sections are obtained, by considering orientations on the various cycles involved. (iii) The orientation-theoretic portion of [Bgg2], Proposition 2.7, referred to in (ii) above may be interpreted, from the point of view of the present paper, as a lifting “C ± I ” of the map C I of Theorem 3.10, (ii), as follows. In the notation of Theorem 3.10, let us write • Cycle n (Π 1 ) ± for the set of pairs consisting of a cycle- subgroup J ∈ Cycle n (Π 1 ) and an orientation on J [cf. Remark 3.6.2, (i)]; • Tpd I (Π 2/1 ) ± for the set of pairs consisting of a tripodal subgroup T ∈ Tpd I (Π 2/1 ) and an orientation on T [cf. Remark 3.6.2, (ii)]. Thus, one has natural surjections Cycle n (Π 1 ) ±  Cycle n (Π 1 ), Tpd I (Π 2/1 ) ±  Tpd I (Π 2/1 ), which may be regarded as torsors over the group {±1}. Moreover, one verifies immediately from the functoriality of the various isomorphisms that appeared in the constructions of Remark 3.6.2, (i), (ii), that the action [cf. Theorem 3.10, (i)] of Aut FC (Π n , I) G on the sets Cycle n (Π 1 ), Tpd I (Π 2/1 ) lifts naturally to an action of Aut FC (Π n , I) G on the sets Cycle n (Π 1 ) ± , Tpd I (Π 2/1 ) ± . Thus, the inverse of the bijective correspondence of the final display of Remark 3.6.2, (ii), determines a natural Aut FC (Π n , I) G -equivariant lift- ing n ± C ± −→ Tpd I (Π 2/1 ) ± I : Cycle (Π 1 ) of the map C I of Theorem 3.10, (ii). [Thus, the Aut FC (Π n , I) G - ± equivariance of C ± I implies, in particular, that C I does not fac- n tor through the natural surjection Cycle (Π 1 ) ±  Cycle n (Π 1 ).] Moreover, if n ≥ 3, and one regards the Π tpd -conjugacy class 114 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI ∗ of cuspidal subgroups of Π tpd determined by J tpd as being “positive”, then it follows immediately from the definition of Tpd I (Π 2/1 ) ± that this lifting C ± I naturally determines an as- signment Cycle n (Π 1 ) ± J ± → syn ± I,J ± — where J ± → J ∈ Cycle n (Π 1 ), and syn ± I,J ± denotes an I- ∼ conjugacy class of isomorphisms Π tpd → C I (J) that coincides, up to possible composition with a cycle symmetry, with the I- conjugacy class of isomorphisms syn I,J of Theorem 3.10, (iii) — such that if, in the diagram [of I tpd -, I-conjugacy classes of isomorphisms] in the display of Theorem 3.10, (iii), (c), one replaces “syn” by “syn ± ”, then the diagram commutes, i.e., even if one does not allow for possible composition with cycle symmetries. Definition 3.11. Suppose that Σ = Primes, and that k = C, i.e., that we are in the situation of Definition 2.22. We shall apply the notational conventions established in Definition 2.22. Moreover, we shall use similar notation def def def def = π 1 (Y E ), Y n = Y {1,...,n} , Y = Y 1 , Y E = (Y E log ) an (C)| s , Y Π disc E Y def disc Y an Y disc Y disc Π disc = Y Π disc p E/E  : Y E → Y E  , Y p Π {1,...,n} , E/E  : Π E  Π E  , n Y def disc Y Π Y disc Π disc E/E  = Ker( p E/E  ) ⊆ Π E , Y an def Y an p n/m = p {1,...,n}/{1,...,m} : Y n −→ Y m , Y Π disc def Y Π disc p n/m = p {1,...,n}/{1,...,m} : Y Π disc  Y Π disc n m , Y Y disc Y disc Y  disc Π (−) , Π disc n/m = Π {1,...,n}/{1,...,m} ⊆ Π n , def Y disc G disc , Y G i∈E,y , Π Y G disc , Π Y G i∈E,y disc for objects associated to the stable log curve Y log = Y 1 log to the notation introduced in Definitions 2.22, 2.23. Definition 3.12. Let J be a semi-graph of temperoids of HSD-type [cf. Definition 2.3, (iii)]. Then we shall refer to a triple ∼ (H, S ⊆ Node(H), φ : H S → J ) [cf. Definition 2.9] consisting of a semi-graph of temperoids of HSD- ∼ type H, a subset S ⊆ Node(H), and an isomorphism φ : H S → J of semi-graphs of temperoids of HSD-type as a degeneration structure on J [cf. [CbTpII], Definition 3.23, (i)]. COMBINATORIAL ANABELIAN TOPICS IV 115 Definition 3.13. In the situation of Definition 3.11: disc ∼ (i) Let ( Y G disc , S ⊆ Node( Y G disc ), φ : Y G S → G disc ) be a degen- eration structure on G disc [cf. Definition 3.12], e ∈ S, and a subgroup of Π disc J ⊆ Π disc 1 1 . Then we shall say that J ⊆ disc Π 1 is a cycle-subgroup of Π disc [with respect to ( Y G disc , S ⊆ 1 ∼ disc Node( Y G disc ), φ : Y G S → G disc ), associated to e ∈ S] if J is disc contained in the Π 1 -conjugacy class of subgroups of Π disc ob- 1 tained by forming the image of a nodal subgroup of Π Y G disc associated to e via the composite of outer isomorphisms Φ −1 Y disc G S ∼ ∼ ∼ disc Π Y G disc −→ Π Y G S disc −→ Π G disc −→ Π 1 — where the first arrow is the inverse of the specialization outer isomorphism Φ Y G S disc [cf. Proposition 2.10], the second ar- row is the graphic [cf. Definition 2.7, (ii)] outer isomorphism ∼ Π Y G S disc → Π G disc induced by φ, and the third arrow is the nat- ∼ of [the second to last ural outer isomorphism Π G disc → Π disc 1 display of] Definition 2.23, (i) [cf. the left-hand portion of Fig- ure 1]. (ii) Let J ⊆ Π 1 disc be a cycle-subgroup of Π disc [cf. (i)]. Thus, we 1 have disc ∼ (a) a degeneration structure ( Y G disc , S ⊆ Node( Y G disc ), φ : Y G S → G disc ) on G disc [cf. Definition 3.12], ∼ (b) an isomorphism Y Π disc → Π disc that is compatible with the 1 1 composite of the display of (i) [cf. also [the second to last display of] Definition 2.23, (i)] in the case where we take disc ∼ the “( Y G disc , S ⊆ Node( Y G disc ), φ : Y G S → G disc )” of (i) to be the degeneration structure of (a), ∼ (c) an isomorphism Y Π disc → Π disc that lifts [cf. Corollary 2.20, 2 2 (v)] the isomorphism of (b) and, moreover, determines a PFC-admissible isomorphism between the respective profi- nite completions, and (d) a nodal subgroup Π e ⊆ Y Π disc of Y Π disc associated to a 1 1 [uniquely determined — cf. Corollary 2.18, (iii)] node e of Y disc G such that the image of the nodal subgroup Π e ⊆ Y Π 1 disc of ∼ → Π disc of (b) coincides with (d) via the isomorphism Y Π disc 1 1 disc . We shall say that a subgroup T ⊆ Π disc J ⊆ Π disc 1 2/1 of Π 2/1 is a tripodal subgroup associated to J if T coincides — rela- tive to some choice of data (a), (b), (c), (d) as above [but cf. also Remark 3.6.1 and Corollary 2.19, (i)!] — with the image, ∼ of (c), of some {1, 2}-tripod in → Π disc via the lifting Y Π disc 2 2 Y disc Π 2/1 ⊆ Y Π disc [cf. Definition 2.23, (ii)] arising from e [cf. Def- 2 inition 2.23, (iii); [CbTpII], Definition 3.7, (i)], and, moreover, 116 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI disc disc the centralizer Z Π disc (T ) maps bijectively, via p Π  2/1 : Π 2 2 disc disc Π 1 , onto J ⊆ Π 1 [cf. Corollary 2.17, (i); [CbTpII], Lemma 3.11, (iv), (vii)]. (iii) Let J ⊆ Π disc be a cycle-subgroup of Π disc [cf. (i)] and T ⊆ Π disc 1 1 2/1 a tripodal subgroup associated to J [cf. (ii)]. Then we shall refer to a subgroup of T that arises from a nodal (respectively, cusp- Y disc idal) subgroup contained in the {1, 2}-tripod in Y Π disc 2/1 ⊆ Π 2 of (ii) as a lifting cycle-subgroup (respectively, distinguished cuspidal subgroup) of T [cf. the right-hand portion of Figure 1]. (iv) Let J ⊆ Π disc be a cycle-subgroup [cf. (i)]; T ⊆ Π disc 1 2/1 a tripodal subgroup associated to J [cf. (ii)]; I ⊆ T a distinguished cuspi- dal subgroup of T [cf. (iii)]. Then it follows immediately from the various definitions involved, together with Theorem 2.24, (i), that there exists a unique outomorphism ι of T such that the induced outomorphism of the profinite completion T  of T coincides with the outomorphism of T  determined by the cycle symmetry of T  associated to the profinite completion I  of I [cf. Definition 3.8]. Moreover, since I is commensurably terminal in T [cf. Corollary 2.18, (v)], it follows immediately from Corollary 2.17, (ii), that there exists a uniquely deter- mined I-conjugacy class of automorphisms of T that lifts ι and preserves I ⊆ T . We shall refer to this I-conjugacy class of automorphisms of T as the cycle symmetry of T associated to I. Theorem 3.14 (Discrete version of canonical liftings of cycles). disc In the notation of Definition 3.11, let I ⊆ Π disc be a cusp- 2/1 ⊆ Π 2 idal inertia group associated to the diagonal cusp of a fiber of p an 2/1 ; disc disc Π tpd ⊆ Π 3 a 3-central {1, 2, 3}-tripod of Π 3 [cf. Definition 2.23, (ii), (iii)]; I tpd ⊆ Π tpd a cuspidal subgroup of Π tpd that does not arise ∗ ∗∗ from a cusp of a fiber of p an 3/2 ; J tpd , J tpd ⊆ Π tpd cuspidal subgroups ∗ ∗∗ , and J tpd determine three distinct Π tpd - of Π tpd such that I tpd , J tpd conjugacy classes of subgroups of Π tpd . [Note that one verifies immedi- ately from the various definitions involved that such cuspidal subgroups ∗ ∗∗ I tpd , J tpd , and J tpd always exist.] For α ∈ Aut FC (Π disc 2 ) [cf. the nota- tional conventions introduced in the statement of Corollary 2.20], write α 1 ∈ Aut FC (Π disc 1 ) determined by α; for the automorphism of Π disc 1 FC disc Aut FC (Π disc 2 , I) ⊆ Aut (Π 2 ) for the subgroup consisting of β ∈ Aut FC (Π disc 2 ) such that β(I) = I; FC G disc Aut FC (Π disc 2 ) ⊆ Aut (Π 2 ) COMBINATORIAL ANABELIAN TOPICS IV 117 for the subgroup consisting of β ∈ Aut FC (Π disc 2 ) such that the image of ∼ FC FC FC disc disc β via the composite Aut (Π 2 )  Out (Π disc 2 ) → Out (Π 1 ) → Out(Π G disc ) — where the second arrow is the natural bijection of Corol- lary 2.20, (v), and the third arrow is the homomorphism induced by ∼ the natural outer isomorphism Π disc → Π G disc — is graphic [cf. Defi- 1 nition 2.7, (ii)]; def FC G disc G Aut FC (Π disc = Aut FC (Π disc 2 , I) 2 , I) ∩ Aut (Π 2 ) ; Cycle(Π disc 1 ) [cf. Definition 3.13, (i)]; for the set of cycle-subgroups of Π disc 1 Tpd I (Π disc 2/1 ) for the set of subgroups T ⊆ Π disc 2/1 such that T is a tripodal subgroup [cf. Definition 3.13, (ii)], associated to some cycle-subgroup of Π disc 1 and, moreover, I is a distinguished cuspidal subgroup [cf. Defini- tion 3.13, (iii)] of T . Then the following hold: G disc disc (i) Let α ∈ Aut FC (Π disc 2 , I) , J ∈ Cycle(Π 1 ), and T ∈ Tpd I (Π 2/1 ). Then it holds that α 1 (J) ∈ Cycle(Π disc 1 ), α(T ) ∈ Tpd I (Π disc 2/1 ). G disc disc Thus, Aut FC (Π disc 2 , I) acts naturally on Cycle(Π 1 ), Tpd I (Π 2/1 ). G (ii) There exists a unique Aut FC (Π disc 2 , I) -equivariant [cf. (i)] map disc C I : Cycle(Π disc 1 ) −→ Tpd I (Π 2/1 ) such that, for every J ∈ Cycle(Π disc 1 ), C I (J) is a tripodal sub- G group associated to J. Moreover, for every α ∈ Aut FC (Π disc 2 , I) ∼ and J ∈ Cycle(Π disc 1 ), the isomorphism C I (J) → C I (α 1 (J)) induced by α maps every lifting cycle-subgroup [cf. Def- inition 3.13, (iii)] of C I (J) bijectively onto a lifting cycle- subgroup of C I (α 1 (J)). (iii) There exists an assignment Cycle(Π disc 1 ) J → syn I,J — where syn I,J denotes an I-conjugacy class of isomorphisms ∼ Π tpd → C I (J) — such that (a) syn I,J maps I tpd bijectively onto I in a fashion that is ∼ compatible with the natural isomorphism I tpd → I in- disc disc  Π disc duced by the projection p Π {1,2,3}/{1,3} : Π 3 {1,3} and ∼ disc the natural outer isomorphism Π disc {1,3} → Π {1,2} obtained by switching the labels “2” and “3” [cf. Corollary 2.17, (ii); Corollary 2.18, (v); [CbTpII], Lemma 3.6, (iv)], ∗ ∗∗ , J tpd bijectively onto lift- (b) syn I,J maps the subgroups J tpd ing cycle-subgroups of C I (J), and 118 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI G (c) for α ∈ Aut FC (Π disc 2 , I) , the diagram [of I tpd -, I-conjugacy classes of isomorphisms] Π tpd −−−→ ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J)  1 C I (J) −−−→ C I (α 1 (J)) — where the upper horizontal arrow is the [uniquely de- termined — cf. the commensurable terminality of I tpd of Π tpd discussed in Corollary 2.18, (v)] I tpd -conjugacy class of automorphisms of Π tpd that lifts T Π tpd (α) [cf. Corol- lary 2.20, (v); Theorem 2.24, (iv)] and preserves I tpd ; the lower horizontal arrow is the I-conjugacy class of isomor- phisms induced by α [cf. (ii)] — commutes up to possible composition with the cycle symmetry of C I (α 1 (J)) as- sociated to I [cf. Definition 3.13, (iv)]. Finally, the assignment J → syn I,J is uniquely determined, up to possible composition with cy- cle symmetries, by these conditions (a), (b), and (c). G (iv) Let α ∈ Aut FC (Π disc 2 , I) and J ∈ Cycle(Π 1 ). Then there ex- G ists an automorphism β ∈ Aut FC (Π disc 2 , I) such that T Π tpd (β) [cf. Corollary 2.20, (v); Theorem 2.24, (iv)] is trivial, and, moreover, α 1 (J) = β 1 (J). Finally, the diagram [of I tpd -, I- conjugacy classes of isomorphisms] Π tpd ⏐ ⏐ syn I,J  Π tpd ⏐ ⏐ syn I,α (J) =syn I,β (J)  1 1 C I (J) −−−→ C I (α 1 (J)) = C I (β 1 (J)) — where the lower horizontal arrow is the isomorphism induced by β [cf. (ii)] — commutes up to possible composition with the cycle symmetry of C I (α 1 (J)) = C I (β 1 (J)) associated to I. Proof. Assertion (i) follows from the various definitions involved. As- sertion (ii) follows immediately from the evident discrete version [cf. Corollaries 2.17, (ii); 2.19, (i)] of the argument involving Remark 3.6.1 that was given in the proof of Theorem 3.10, (ii). The existence portion of assertion (iii) follows, in light of Corollaries 2.17, (ii); 2.20, (i), (v), from a similar argument to the argument applied in the proof of the ex- istence portion of Theorem 3.10, (iii) [cf. also the fact that the “syn I,J ” of Theorem 3.10, (iii), was constructed from a suitable geometric outer isomorphism]. The uniqueness portion of assertion (iii) follows from COMBINATORIAL ANABELIAN TOPICS IV 119 the compatibility portion of condition (a), together with the compu- tation of discrete outomorphism groups given in Theorem 2.24, (ii). Assertion (iv) follows immediately from assertion (iii), together with a similar argument to the argument applied in the proof of the surjectiv- ity portion of Theorem 2.24, (iv) [cf. the argument given in the proof of Theorem 3.10, (iv)]. This completes the proof of Theorem 3.14.  Remark 3.14.1. One verifies immediately that the discrete construc- tions of Theorem 3.14, (i), (ii), (iii), (iv), are compatible, in an evident sense, with the pro-Σ constructions of Theorem 3.10, (i), (ii), (iii), (iv). We leave the routine details to the reader. Remark 3.14.2. One verifies immediately that remarks analogous to Remarks 3.6.2, 3.10.1 in the profinite case may be made in the dis- crete situation treated in Theorem 3.14. In this context, we observe that the theory of the “modules of local orientations Λ” developed in [CbTpI], §3, admits a straightforward discrete analogue, which may be ∼ applied to conclude that the “orientation isomorphisms J → Λ G ” of Remark 3.6.2, (i), are compatible with the natural discrete structures on the domain and codomain. Alternatively, in the discrete case, relative to the notation of Definition 2.2, (iii), one may think of these modules “Λ” as the Z-duals of the second relative singular cohomology modules [with Z-coefficients] H 2 (U X , ∂U X ; Z) — cf. the discussion of orientations in [CbTpI], Introduction. Then the discrete version of the key isomorphisms [cf. the constructions of Remark 3.6.2] of [CbTpI], Corollary 3.9, (v), (vi), may be obtained by considering the connecting homomorphism [from first to second coho- mology modules] in the long exact cohomology sequence associated to the pair (U X , ∂U X ). We leave the routine details to the reader. 120 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI Appendix. Explicit limit seminorms associated to sequences of toric surfaces In the proof of Corollary 1.15, (ii), we considered sequences of dis- crete valuations that arose from vertices or edges of the dual semi- graphs associated to the geometric special fibers of a tower of coverings of stable log curves and, in particular, observed that the convergence of a suitable subsequence of such a sequence follows immediately from the general theory of Berkovich spaces. In the present Appendix, we reexamine this convergence phenomenon from a more elementary and explicit — albeit logically unnecessary, from the point of view of proving Corollary 1.15, (ii)! — point of view that only requires a knowledge of elementary facts concerning log regular log schemes, i.e., without applying the terminology and notions [e.g., of “Stone-Čech compact- ifications”] that frequently appear in the general theory of Berkovich spaces [cf. the proof of [Brk1], Theorem 1.2.1]. In particular, we discuss the notion of a “stratum” of a “toric surface” [cf. Definition A.1 below], which generalizes the notion of a vertex or edge of the dual graph of the special fiber of a stable curve over a complete discrete valuation ring. We observe that such a stratum determines a discrete valuation [cf. Definition A.4] and consider, at a quite explicit level, the limit of a suitable subsequence of a given sequence of such discrete valuations [cf. Theorem A.7 below]. The material presented in this Appendix is quite elementary and “well-known”, but we chose to include it in the present paper since we were unable to find a suitable reference that discusses this material from a similar point of view. In the present Appendix, let R be a complete discrete valuation ring. Write K for the field of fractions of R and S log for the log scheme def obtained by equipping S = Spec(R) with the log structure determined by the unique closed point of S. Definition A..1. (i) We shall refer to an fs log scheme X log over S log as a toric surface over S log if the following conditions are satisfied: (a) The underlying scheme X of X log is of finite type, flat, and of pure relative dimension one [i.e., every irreducible component of every fiber of the underlying morphism of schemes X → S is of dimension one] over S. (b) The fs log scheme X log is log regular. (c) The interior [cf., e.g., [MT], Definition 5.1, (i)] of the log scheme X log is equal to the open subscheme X × R K ⊆ X . Given two toric surfaces over S log , there is an evident notion of isomorphism of toric surfaces over S log . (ii) Let X log be a toric surface over S log [cf. (i)] and n a nonnegative integer. Write X [n] ⊆ X for the n-interior of X log [cf. [MT], Definition 5.1, (i)] and X [−1] ⊆ X for the empty subscheme. COMBINATORIAL ANABELIAN TOPICS IV 121 Then we shall refer to a connected component of X [n] \ X [n−1] as an n-stratum of X log . We shall write Str n (X log ) for the set of n-strata of X log [so Str n (X log ) = ∅ if n ≥ 3] and def Str(X log ) = Str 1 (X log )  Str 2 (X log ). Definition A..2. Let I be a totally ordered set that is isomorphic to N [equipped with its usual ordering]. In particular, it makes sense to speak of “limits i → ∞” of collections of objects indexed by i ∈ I, as well as to speak of the “next largest element” i + 1 ∈ I associated to a given element i ∈ I. Then we shall refer to a sequence of fs log schemes log · · · −−−→ X i+1 −−−→ X i log −−−→ · · · — where i ranges over the elements of I — over S log [indexed by I] as a sequence of toric surfaces over S log if, for each i ∈ I, X i log is a toric surface over S log [cf. Definition A.1, (i)], and, moreover, the morphism log → X i log is dominant. Observe that the horizontal arrows of the X i+1 above diagram determine [by considering the induced maps of generic points of strata] a sequence of maps of sets log ) −→ Str(X i log ) −→ · · · . · · · −→ Str(X i+1 Finally, given two sequences of toric surfaces over S log , there is an evident notion of isomorphism of sequences of toric surfaces over S log . Definition A..3. Let X log be a toric surface over S log and A a strict henselization of X at [the closed point determined by] z ∈ Str 2 (X log ) [cf. Definition A.1, (i), (ii)]. Write F for the field of fractions of A; k for def the residue field of A; m A for the maximal ideal of A; X z = Spec(A); M X for the sheaf of monoids on X that defines the log structure of X log ; M for the fiber of M X /O X × at the maximal ideal of A; def def def Q = Hom(M, Q ≥0 ) ⊆ P = Hom(M, R ≥0 ) ⊆ V = Hom(M, R) — where we write Q ≥0 , R ≥0 for the respective submonoids deter- mined by the nonnegative elements of the [additive groups] Q, R and “Hom(M, −)” for the monoid consisting of homomorphisms of monoids from M to “(−)”. Thus, one verifies easily that V is equipped with a natural structure of two-dimensional vector space over R. In the fol- lowing, we shall use the superscript “gp” to denote the groupification of any of the monoids of the above discussion. (i) We shall say that a submonoid L ⊆ P of P is a P -ray if L is the R ≥0 -orbit of some nonzero element of P , relative to the natural [multiplicative] action of R ≥0 on P . 122 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (ii) We shall say that a P -ray L ⊆ P [cf. (i)] is rational (respec- tively, irrational) if L ∩ Q  = {0} (respectively, L ∩ Q = {0}). (iii) Let L ⊆ P be a rational P -ray [cf. (i), (ii)]. Then we shall write v L : F × → Q ⊆ R for the discrete valuation associated to the irreducible component of the blow-up of X z associated to L ⊆ P , normalized so as to map each prime element π R of R ⊆ F to 1 ∈ Q. That is to say, if λ ∈ L [which, by a slight abuse of notation, we regard as a homomorphism M gp → R] maps π R → 1 ∈ Q [so λ ∈ L ∩ Q], and f ∈ F lies in the A × -orbit determined by m ∈ M gp , then v L (f ) = λ(m) ∈ Q. Here, we observe that [one verifies easily that] the submonoid def M L = λ −1 (Q ≥0 ) ⊆ M gp is isomorphic to Z × N. In particular, if we denote by F L ⊆ F the set of f ∈ F that lie in the A × -orbits determined by m ∈ M L and write A L ⊆ F for the A-subalgebra generated by f ∈ F L , then the “blow-up of X z associated to L” referred to above may be described explicitly as def X L = Spec(A L ) −→ X z . Indeed, if we write p L ⊆ A L for the ideal generated by the set of f ∈ F that lie in the A × -orbits determined by the noninvertible elements m ∈ M L , then it follows immediately from the simple structure of the monoid Z × N that p L is the prime ideal of height one in A L that corresponds to the discrete valuation v L , and that the k-algebra A L /p L is isomorphic to k[U, U −1 ], where U is an indeterminate. (iv) Write M S for the sheaf of monoids on S that defines the log structure of S log ; M R for the fiber of M S /O S × at the unique def closed point of S; V R = Hom(M R , R). Then one verifies easily that V R is a one-dimensional vector space over R, and that the morphism X log → S log determines an R-linear surjection V  V R . Let e α , e β ∈ P be such that R ≥0 · e α + R ≥0 · e β = P , and, moreover, the images of e α , e β in V R coincide. [Note that the existence of such elements e α , e β ∈ P follows, e.g., from [ExtFam], Proposition 1.7.] Then we shall refer to the [necessarily rational — cf. (ii)] P -ray R ≥0 · (e α + e β ) ⊆ P [cf. (i)] as the midpoint P -ray at z ∈ Str 2 (X log ). Here, we note that one verifies easily that the P -ray R ≥0 · (e α + e β ) does not depend on the choice of the pair (e α , e β ). (v) We shall refer to a valuation w : F × → R as admissible if w dominates A and maps each prime element π R of R ⊆ F to 1 ∈ R. Let w be an admissible valuation. Then by restricting w to the elements f ∈ F that lie in the A × -orbits determined COMBINATORIAL ANABELIAN TOPICS IV 123 by m ∈ M , one obtains a nonzero homomorphism of monoids M → R ≥0 , i.e., an element of P . We shall refer to the P -ray L w determined by this element of P as the P -ray associated to the admissible valuation w. Thus, if L w is rational [cf. (ii)], then it follows immediately from the definitions that, in the notation of (iii), the valuation of A determined by w extends to a valuation of A L w (⊇ A). Remark A..3.1. In the notation of Definition A.3, the usual topology on the real vector space V naturally determines a topology on the subspace P ⊆ V , as well as on the set of P -rays [i.e., which may be regarded as the complement of the “zero element” in the quotient space P/R ≥0 ]. Moreover, one verifies easily that, if e α and e β are as in Definition A.3, (iv), then the assignment R ⊇ [0, 1] γ → R ≥0 · (γ · e α + (1 − γ) · e β ) determines a homeomorphism of the closed interval [0, 1] ⊆ R onto the resulting topological space of P -rays, and that the subset of rational P -rays is dense in the space of P -rays. In particular, it makes sense to speak of non-extremal (respectively, extremal) P -rays, i.e., P -rays that lie (respectively, do not lie) in the interior — i.e., relative to the homeomorphism just discussed, the open interval (0, 1) ⊆ [0, 1] (respectively, the endpoints {0, 1} ⊆ [0, 1]) — of the space of P -rays. Finally, we observe that the two extremal P -rays are rational, and that a rational P -ray is non-extremal if and only if its associated discrete valuation [cf. Definition A.3, (iii)] is admissible [cf. Definition A.3, (v)]. Definition A..4. Let X log be a toric surface over S log ; z ∈ Str(X log ) [cf. Definition A.1, (i), (ii)]. Write F for the residue field of the generic point of the irreducible component of X on which [the subset of X determined by] z ∈ Str(X log ) lies. Then one may associate to z ∈ Str(X log ) a collection of distinguished valuations on F , as well as a uniquely determined canonical valuation on F , as follows: (i) If z is a 1-stratum, then we take both the unique distinguished valuation and the canonical valuation associated to z to be the discrete valuation F × −→ Q ⊆ R associated to the prime of height 1 determined by z, normalized so as to map each prime element π R of R ⊆ F to 1 ∈ Q. (ii) If z is a 2-stratum, then we take the collection of distinguished valuations associated to z to be the discrete valuations F × −→ Q ⊆ R 124 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI determined by the restrictions of the discrete valuations asso- ciated to the rational P -rays [cf. Definition A.3, (iii)]. We take the canonical valuation associated to z to be the discrete val- uation determined by the restriction of the discrete valuation associated to the midpoint P -ray at z [cf. Definition A.3, (iii), (iv)]. Here, we note that the construction from z of either the collection of dis- tinguished valuations or the uniquely determined canonical valuation is functorial with respect to arbitrary isomorphisms of pairs (X log , z) [i.e., pairs consisting of a toric surface over S log and an element of “Str(−)” of the toric surface]. Remark A..4.1. One verifies immediately that the [noncuspidal] val- uations of the discussion preceding Corollary 1.15 correspond precisely to the canonical valuations of Definition A.4. Lemma A..5 (Valuations associated to irrational rays). In the notation of Definition A.3, let L ⊆ P be an irrational P -ray [cf. Definition A.3, (i), (ii)], {L i } ∞ i=1 a sequence of P -rays such that L = lim i→∞ L i [cf. Remark A..3.1], and {w i } ∞ i=1 a sequence of admissible valuations such that, for each positive integer i, L i is the P -ray asso- ciated to w i [cf. Definition A.3, (v)]. Then there exists an admissible valuation [cf. Definition A.3, (v)] v L : F × −→ R which satisfies the following conditions: (a) The P -ray associated to v L [cf. Definition A.3, (v)] is equal to L. (b) For each f ∈ F × , it holds that v L (f ) = lim w i (f ). i→∞ (c) If λ ∈ L maps a prime element π R of R to 1 ∈ R, J is a nonempty finite set, {m j } j∈J is a collection of distinct ele- ments of M gp , and {f j } j∈J is a collection of elements of F such that f j lies in the A × -orbit determined by m j , then  v L ( f j ) = min λ(m j ) ∈ R. j∈J j∈J Moreover, this valuation v L is the unique admissible valuation [i.e., in the sense of Definition A.3, (v)] that satisfies condition (a). In partic- ular, v L depends only on the P -ray L ⊆ P , i.e., is independent of ∞ the choice of the sequences {L i } ∞ i=1 and {w i } i=1 . COMBINATORIAL ANABELIAN TOPICS IV 125 Proof. One may define a map v L : F × → R by applying the formula in the display of condition (c) in the case where J  = 1. Then one verifies easily that this map v L is a homomorphism [with respect to the multiplicative structure of F × ] and satisfies condition (b). Next, let us observe that since [we have assumed that] L is irrational, the map M gp → R determined by λ ∈ L is injective. Thus, it follows from condition (b), together with the fact that each of the w i ’s is a valuation, that the map v L satisfies condition (c), which implies that the map v L is a [necessarily admissible] valuation on F . Moreover, it follows immediately from the definition of v L that v L satisfies condition (a). This completes the proof of Lemma A.5.  Lemma A..6 (Convergence of midpoints of closed intervals). Let def · · · ⊆ [a i+1 , b i+1 ] ⊆ [a i , b i ] ⊆ [a i−1 , b i−1 ] ⊆ · · · ⊆ [a 0 , b 0 ] = [0, 1] ⊆ R — where i ranges over the nonnegative integers — be a sequence of inclusions of nonempty closed intervals in [0, 1]. For each i, write c i for def the midpoint of the closed interval [a i , b i ], i.e., c i = (a i +b i )/2 ∈ [a i , b i ]. Then the sequence of midpoints {c i } ∞ i=1 converges. Proof. This follows immediately from the [easily verified] fact that the ∞  sequences {a i } ∞ i=1 , {b i } i=1 converge. Theorem A..7 (Explicit limit seminorms associated to sequences of toric surfaces). Let R be a complete discrete valuation ring and I a totally ordered set that is isomorphic to N [equipped with its usual ordering]. Write K for the field of fractions of R and S log for the def log scheme obtained by equipping S = Spec(R) with the log structure determined by the unique closed point of S. Let log · · · −−−→ X i+1 −−−→ X i log −−−→ · · · be a sequence of toric surfaces over S log indexed by I [cf. Definition A.2] and {z i } i∈I ∈ lim Str(X i log ) ←− i∈I [cf. Definitions A.1, (ii); A.2]. Then, after possibly replacing I by a suitable cofinal subset of I, there exist sequences {v i : F i × → R} i∈I , {v z i } i∈I — where, for each i ∈ I, F i denotes the residue field of some point x i ∈ X i × R K; v i : F i × → R is a valuation; v z i is a distinguished valuation associated to z i [cf. Definition A.4] — such that 126 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI (a) v i maps each prime element of R ⊆ F i to 1 ∈ R [which thus implies that v i dominates R]; (b) the x i ’s and v i ’s are compatible [in the evident sense] with log → X i log of the above respect to the upper horizontal arrows X i+1 diagram; (c) for every nonzero rational function f on the irreducible compo- nent of X i containing x i that is regular at x i , hence determines an element f ∈ F i [cf. Remark A..7.1 below], it holds that v i (f ) = lim v z j (f ) j→∞ [cf. Definition A.4] — where j ranges over the elements of I that are ≥ i, and we regard v i as a map defined on F i by sending F i 0 → +∞. Finally, these sequences of valuations {v i } i∈I , {v z i } i∈I may be con- structed in a way that is functorial [in the evident sense] with respect to isomorphisms of pairs consisting of a sequence of toric surfaces over S log and a compatible collection of strata [i.e., “{z i } i∈I ”]. Proof. Until further notice, we take, for each i ∈ I, v z i to be the canon- ical valuation associated to z i [cf. Definition A.4]. Next, let us observe that one verifies easily that we may assume without loss of generality, by replacing I by a suitable cofinal subset of I, that there exists an element n ∈ {1, 2} such that every member of {z i } is an n-stratum, i.e., one of the following conditions is satisfied: (1) Every member of {z i } is a 1-stratum. (2) Every member of {z i } is a 2-stratum. First, we consider Theorem A.7 in the case where condition (1) is satisfied. For each i ∈ I, write Z i ⊆ X i for the reduced closed sub- scheme of X i whose underlying closed subset [⊆ X i ] is the closure of the subset of X determined by the 1-stratum z i . Then let us observe that if, after possibly replacing I by a suitable cofinal subset of I, it holds that, for each i ∈ I, the composite Z i+1 → X i+1 → X i is quasi-finite, then the system consisting of the v z i ’s [cf. Definition A.4, (i)] already yields a system of valuations {v i } i∈I as desired. Thus, we may assume without loss of generality, by replacing I by a suitable cofinal subset of I, that, for each i ∈ I, the composite Z i+1 → X i+1 → X i is not quasi- finite, i.e., that the image of this composite is a closed point y i ∈ X i of X i . Here, we observe that since we are operating under the assumption that condition (1) is satisfied, it follows from the fact that z i+1 → z i that y i necessarily lies in the regular locus of X i . For each i ∈ I, write B i for the local ring of X i at y i ∈ X i , E i for the field of fractions of B i , and v z i : E i × → R for the discrete valuation defined in Definition A.4, (i). Thus, one verifies immediately that the morphisms · · · → X i+1 → X i → · · · COMBINATORIAL ANABELIAN TOPICS IV 127 induce compatible chains of injections · · · → B i → B i+1 → · · · , · · · → E i → E i+1 → · · · . Moreover, if π R is a prime element of R, then the discrete valuation v z i may be interpreted as the discrete valuation of B i determined by the unique height one prime of B i that contains π R . In particular, since B i is regular, hence a unique factorization domain, one verifies immediately — by considering the extent to which positive powers of an element f ∈ B i are divisible, in B i or in B i+1 , by positive powers of π R — that, for each i ∈ I and f ∈ B i , it holds that (0 ≤) v z i (f ) ≤ v z i+1 (f ). (∗) For each i ∈ I, write def p i = { f ∈ B i | lim v z j (f ) = +∞ } ⊆ B i . j→∞ Then since each v z j is a [discrete] valuation, one verifies immediately that p i ⊆ B i is a prime ideal of B i . Moreover, since π R ∈ p i , we conclude that the ideal p i is not maximal, i.e., that the height of p i is ∈ {0, 1}. Next, let us observe that if, after possibly replacing I by a suitable cofinal subset of I, it holds that, for each i ∈ I, the prime ideal p i is of height 1, then it follows immediately that p i determines a closed point x i of the generic fiber of X i , and that, if we write F i for the residue field of X i at x i and v i : F i × → R for the uniquely determined [since F i is a finite extension of K] discrete valuation on F i that extends the given discrete valuation on K and maps π R → 1 ∈ R, then the limit lim j→∞ v z j (−) [cf. (∗)] determines a valuation on F i = (B i ) p i /p i (B i ) p i that necessarily coincides [since F i is a finite extension of K] with v i ; in particular, one obtains a system of valuations {v i } i∈I as desired. Thus, we may assume without loss of generality, by replacing I by a suitable cofinal subset of I, that, for each i ∈ I, the prime ideal p i is of height 0, i.e., p i = {0}, hence determines a generic point x i of some irreducible component of X i such that E i may be naturally identified with the residue field F i of X i at x i . But this implies that, for f ∈ E i × = F i × , the quantity def v i (f ) = lim v z j (f ) ∈ R j→∞ is well-defined [cf. (∗)]. Moreover, one verifies immediately that this definition of v i determines a valuation on E i = F i . In particular, one obtains a system of valuations {v i } i∈I as desired. This completes the proof of Theorem A.7 in the case where condition (1) is satisfied. Next, we consider Theorem A.7 in the case where condition (2) is satisfied. For each i ∈ I, write Q i , P i , V i for the objects “Q”, “P ”, “V ” defined in Definition A.3 in the case where we take the data “(X log , z ∈ Str 2 (X log ))” in Definition A.3 to be (X i log , z i ∈ Str 2 (X i log )). Then one 128 YUICHIRO HOSHI AND SHINICHI MOCHIZUKI log verifies easily that the morphism X i+1 → X i log determines a nontrivial R-linear map V i+1 → V i that maps Q i+1 , P i+1 ⊆ V i+1 into Q i , P i ⊆ V i , respectively. Next, let us observe that if, after possibly replacing I by a suitable cofinal subset of I, it holds that, for each i ∈ I, the R-linear map V i+1 → V i is of rank one, i.e., the image of P i+1 ⊆ V i+1 in V i is a rational P i -ray L i [cf. Definition A.3, (i), (ii)], then we may assume without loss of generality, by taking v z i to be the distinguished valuation associated to the rational P i -ray L i [cf. Definition A.4, (ii); Remark A..7.2 below] and then replacing the pair (X i , z i ) by the pair consisting of the blow- up of X i and the 1-stratum of this blow-up determined by L i [cf. the discussion of Definition A.3, (iii)], that condition (1) is satisfied. Thus, we may assume without loss of generality, by replacing I by a suitable cofinal subset of I, that, for each i ∈ I, the R-linear map V i+1 → V i is of rank  = 1, hence [cf. the existence of the R-linear surjection “V  V R ” of Definition A.3, (iv)] of rank two, i.e., an isomorphism. Since the R-linear map V i+1 → V i is an isomorphism, it follows im- mediately from Lemma A.6, together with Remark A..3.1, that, for each i ∈ I, the sequence consisting of the images in P i of the midpoint P j - rays [cf. Definition A.3, (iv)], where j ranges over the elements of I such that j ≥ i, converges to a [not necessarily rational] P i -ray L i,∞ ⊆ P i . If, after possibly replacing I by a suitable cofinal subset of I, it holds that, for each i ∈ I, the P i -ray L i,∞ is rational, then we may assume without loss of generality, by taking v z i to be the distinguished valua- tion associated to the rational P i -ray L i,∞ [cf. Definition A.4, (ii); Remark A..7.2 below] and then replacing the pair (X i , z i ) by the pair consisting of the blow-up of X i and the 1-stratum of this blow-up deter- mined by L i,∞ [cf. the discussion of Definition A.3, (iii)], that condition (1) is satisfied. Thus, it remains to consider the case in which we may assume without loss of generality, by replacing I by a suitable cofinal subset of I, that, for each i ∈ I, the P i -ray L i,∞ is irrational. Then the system consisting of the valuations v L i,∞ ’s of Lemma A.5 yields a system of valuations {v i } i∈I as desired. This completes the proof of Theorem A.7.  Remark A..7.1. In the situation of Theorem A.7, for I j ≥ i, write z j i for the irreducible locally closed subset of X i determined by the image of the stratum z j in X i . Thus, z j i  ⊆ z j i for all j  ≥ j, and one verifies immediately that the intersection  def i z ∞ = z j i j≥i is nonempty. Moreover, it follows immediately from the constructions i discussed in the proof of Theorem A.7 that if ξ i ∈ z ∞ , then any element COMBINATORIAL ANABELIAN TOPICS IV 129 f of the local ring O X i ,ξ i of X i at ξ i determines a rational function on the irreducible component of X i containing x i that is regular at x i [cf. Theorem A.7, (c)]. Remark A..7.2. 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(Yuichiro Hoshi) Research Institute for Mathematical Sciences, Ky- oto University, Kyoto 606-8502, JAPAN Email address: yuichiro@kurims.kyoto-u.ac.jp (Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN Email address: motizuki@kurims.kyoto-u.ac.jp